text:
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Holmes Family Newsletter Vol 4 No 3
by bill - 1992-03-20 ( family / holmes / newsletter )
The Leaky Faucet
Vol. 4, No. 3, March 18, 1992
"More of the same, basically."
Doug elected president
With this nation in the throes of its quadrennial tradition of political rhetoric and b.s., one name has gone relatively unnoticed: Douglas Phillip [redacted]. Never heard of him? That will soon change. You see, Mr. [redacted] has been elected President of the Sacramento Chapter of the Hungarian/American Friendship Society. And with the considerable power and persuasion that comes with the title, we will all soon be feeling Doug's presence.
We talked to Doug, and here is what he had to say:
LF: What sort of sweeping changes will we be seeing in the near future, now that you've
been elected President?
Doug: First of all, everyone will be required to speak Hungarian.
LF: Hungarian? What if they don't want to?
Doug: Well, too bad!
LF: Okay, fine. Hungarian. What else can we expect in the future?
Doug: Everyone will be required to drink coffee out of personalized mugs made
exclusively by Doug's Mugs.
LF: But you don't drink coffee. What about people like yourself? Will they be required
to start drinking coffee?
Doug: I'm afraid so. It's for the good of The Cause.
LF: What "cause" is that?
Doug: I'm trying to save up enough money to make a second trip to Hungary. The only way
I'll be able to do that is if I sell enough mugs.
LF: A noble cause, indeed. How about the other social ills of this nation? You know,
unemployment, homelessness, crime, the environment, bad television. How will you address these problems?
Doug: Just buy my mugs and everything else will take care of itself.
Dad & Eleanor visit Arizona
Their trip began in Sacramento (because that's where they live). They took Highway 395 south through Bishop, CA; through Las Vegas; and, finally, into the enchanted lands of Phoenix, Arizona (where they attended an ancient ritual know as a "wedding"). After the wedding, Dad and Eleanor went to Jon & Cathi's (the bride's parents) house in Cave Creek, AZ. They saw no caves. No creek. But they did get a tour of the backyard. (And this time, Dad refrained from killing any rattlesnakes.)
From Phoenix, Dad & Eleanor headed west on Interstate 8 through Yuma. In eastern San Diego County the highway ran through mountains of boulders the size of a car! It was hell on the car's suspension.
Finally, they entered the enchanted lands of Santa Ana, where they had dinner with Greg, June and family. Aileen played the piano, Brian drew some cartoons (see below), and Andy - well, Andy was Andy.
After leaving Santa Ana, the car's alternator went out right in the middle of the morning rush-hour traffic. But they retrieved it and, $167 later, they were back on the freeway and heading north to Sacramento.
On the return trip from Phoenix to Sacramento, Dad & Eleanor's Mercury Cougar averaged 36 miles per gallon (!) and brought the odometer to 142,000 miles (for those of you keeping track).
Bill Gets Perm Job
Bill has taken a position as an administrative assistant in the immigration department of the "world's largest law firm," Baker & McKenzie (they have 50 offices worldwide).
As you can see from the photo, Bill likes to wear a hard hat and walk around with a walkie-talkie.
We're now taking bets on how long he'll keep this job.
M I S C E L L A N Y
A cheerful thought: According to Nostradamus (or, at least, one person's interpretation of the 16th Century astrologer's predictions), "The Big One" [earthquake] will hit California on May 8, 1993. It is said San Diego will sink into the ocean; Los Angeles will be destroyed; and, by way of the San Francisco Bay, the entire California valley floor will be flooded (completely ruining California's farmlands, and everybody's day).
Other than that, 1993 should be a good year.
There was an interesting article in the L.A. Times the other day. It began: "Times are so lean in the north country [Montana] this winter that ... its citizens are fighting over road-kill." The article goes on to say that "Most Montanans look at the road-kill and say, 'Gee, that's a good piece of meat. It shouldn't go to waste.'" Also, "A couple hundred pounds of meat lying by the roadside is all the more appealing during these recessionary times."
Yum, I'm getting hungry.
There was a movie on t.v. recently called "Cold Steel." You can guess by the title that it was your typically predictable "crime drama." Actually, it wasn't the movie that was noteworthy, but the theme song. Then again, it wasn't so much the song itself as it was who sang the song. Well, by now you've probably guessed it. Yes, that's right, the singer was none other than James House, Don's old crony! Of course, it was recorded in James's pre-Nashville days.
Stormy Gets Schutzhund I
Lucy's dog, Stormy, aka "Dummy," aka "Hey You," has achieved something called "Schutzhund I." It's basically a training certificate awarded to dogs that can: (1) track down a scented object; (2) attack a scented human; and (3) obey a scented handler's commands. Of course, with all these scents, we asked Lucy, "What scent attracts Stormy the most?" Lucy replied, "Well, the smell of dog sht probably attracts him the most. But we try to discourage its use in the Schutzhund trials."
BIRTHDAYS, ETC.
March 17 St. Patrick's Day
April 1 April Fool's Day
April 2 Dad
April 20 Greg
April 21 Earth Day
AN UNPAID ADVERTISEMENT
Because he has an 800 number (1-800-426-1112). Besides, he has some seriously good ideas; namely, his flat 13% tax proposal, plus the fact that he's not afraid to blow the lid off the corruption currently infesting Capitol Hill. Of course, he'll probably lose to Bill Clinton and Clinton will lose to George Bush. Then again, maybe not.
What's going on here? A couple of issues back, this newsletter predicted that a Democrat (but, specifically not Jerry Brown) would be elected President in November. And now, here you are plugging Jerry Brown!
So what's your point?
WHATEVER HAPPENED TO ... Burton Cummings
Formerly of the Canadian rock group "The Guess Who," as well as a solo artist, Burton Cummings is now doing the Canadian and American national anthems at L.A. Kings hockey games.
It's sad, really.
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Q: navbar toggle button shifts right on click i have a bootstrap navbar on m webpage and it has a toggle button which is on right side. But when i click on it, it shifts right by 10px and then when it collapses, it shifts back. This looks weird. I tried margin: 0 auto !important; in css. But this makes toggle button's position center. I want my toggle button to appear on right and should not move on click.
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A: Add this class CSS into your code.
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\section{Introduction}
Let $X$ be a smooth projective surface over a field $\bfk$ whose anti-canonical divisor $-K_X$ is big. Although the classification of such surfaces is completely understood if $- K _{ X }$ is also nef, the question becomes much harder if we drop the nefness from the assumption. If $X$ is rational, then it is known to be a Mori dream space as is shown in \cite[Theorem 1]{MR2824848}. They also gave a kind of structure theorem for such surfaces \cite[Theorem 2]{MR2824848}.

Let us consider the case when $X$ is not rational.
Since the Kodaira dimension of $X$ is $-\infty$, $X$ is obtained by repeatedly blowing up a geometrically ruled surface, whose anti-canonical line bundle is easily seen to be big.
In the first part of this paper, we study the structure of geometrically ruled surfaces whose anti-canonical divisor is big.
We will show the following theorem.

\begin{theorem}[$=$\pref{th:main_theorem}]\label{th:int1}
Let
$
 C
$
be a smooth projective curve of genus $g \ge 1$ and
$
 E
$
be an unstable vector bundle of rank $2$ on $C$. Set
$
 X \coloneqq \bP _{ C } (E)
$.
Then $-K_X$ is big if and only if
$
 E \simeq L \oplus M
$
for some line bundles $L$ and $M$ on $C$ such that
$
 \deg L- \deg M > 2 g - 2
$.
\end{theorem}

We first show, as \pref{cr:main_theorem_split_case}, \pref{th:int1} under the assumption that $E$ is decomposable.
In general we consider a degeneration of $E$ to the direct sum of its Harder-Narasimhan factors and then apply \pref{cr:main_theorem_split_case} to it.
To conclude the proof, we \emph{a posteriori} show that $E$ in fact is a direct sum of its Harder-Narasimhan factors.

The instability of $E$, which we assumed in \pref{th:int1}, follows almost automatically from the bigness of $- K _{ X }$.

\begin{theorem}[$=$\pref{cr:sss_implies_big}]\label{th:int2}
Let $C$ be as in \pref{th:int1}, and $E$ be a vector bundle of rank $r$
on $C$ such that
$
 - K _{ \bP _{ C } \lb E \rb }
$
is big. Then $E$ is not strongly semi-stable; namely, the pull-back of $E$
by an iteration of the Frobenius morphism of $C$ is unstable.
In particular,
$E$ itself is unstable if
$
 \operatorname{char} ( \bfk ) = 0
$
or $ g = 1 $.
\end{theorem}

An example of a curve of genus $g \ge 2$ and a rank $2$
semi-stable vector bundle on it which is not strongly semi-stable is constructed in \cite[Theorem 1]{MR0325616}.
The anti-canonical divisor of the projective bundle associated to the example, however, does not
seem to be big.
Hence the following subtle question remains open.

\begin{question}
Is $E$ itself unstable under the assumption of \pref{th:int2}?
\end{question}

Summing up the two results above, we obtain the following corollary.

\begin{corollary}\label{cr:main_cor}
\
\begin{enumerate}
\item
Suppose either
$
 \operatorname{char} ( \bfk ) = 0
$
or $ g = 1 $. Then
$
 - K _{ X }
$
is big if and only if
$
 E \simeq L \oplus M
$
for some line bundles $L$ and $M$ on $C$ such that
$
 \deg L - \deg M > 2 g - 2
$.

\item
Suppose that
$
 \operatorname{char} ( \bfk ) > 0
$
and
$
 g > 1.
$
If
$
 - K _{ X }
$
is big, then there exists
$
 e \ge 0
$
such that
$
 \lb F ^{ e } \rb ^{ * } E \simeq L ' \oplus M '
$
for some line bundles $L '$ and $M '$ on $C$ such that
$
 \deg L ' - \deg M ' > 2 g - 2
$,
where $F$ is the Frobenius map of $C$.
\end{enumerate}
\end{corollary}

The second aim of this paper is to show the following version of the base point free theorem as an application of \pref{cr:main_cor}.

\begin{theorem}[$=$\pref{cr:adj}]\label{th:adj}
Let $X$ be a normal projective surface over a field $\bfk$ such that
$
 - ( K_X + \Delta )
$
is a nef and big $\bR$-Cartier divisor for some
$\bR$-divisor $\Delta$ such that $\lfloor \Delta \rfloor = 0$.
Then $\Pic ( X )$ is a free abelian group of finite rank.
\end{theorem}

The base point free theorem on surfaces in positive characteristics is shown in \cite[Theorem 3.2]{MR3383599} for numerically non-trivial nef divisors. \pref{th:adj} is nothing but the base point free theorem for numerically trivial divisors, and it is shown in \cite[Corollary 3.6]{MR3383599} under the assumption that $X$ admits only rational singularities. In this case the surface $X$ is necessarily $\bQ$-factorial, so that one can use the minimal model theory. A typical example which is not covered is the cone over a non-singular plane cubic curve, but by \pref{th:adj}, the base point theorem for surfaces is completely settled.

In fact, however, \pref{th:adj} for the case when $X$ admits a non-rational singularity is covered by \cite[Theorem 2.2 (iii)]{MR1821186}. Hence,
as discussed in \pref{sc:combination} below,
we can also show \pref{th:adj} by combining these results.

It is interesting to observe what happens if we allow
$
 \Delta
$
to take coefficients 1.
If we assume that
$
 - \lb K _{ X } + \Delta \rb
$
is ample, then our proof based on \pref{cr:main_cor} works equally well.
However, the assertion of \pref{th:adj} does not necessarily hold when
$
 - \lb K _{ X } + \Delta \rb
$
is assumed to be only nef and big (see \pref{eg:counter}).
In \pref{sc:Delta=1} we investigate when the assertion of \pref{th:adj} fails,
by closely examining the proof of \pref{cr:adj}. For example, it is shown that $X$ needs to be birationally equivalent to the product of $\bP ^{ 1 }$ and a curve of genus one for the failure.

In the last section, we construct some examples of non-minimal ruled surfaces with big anti-canonical divisor by blowing-up geometrically ruled surfaces along certain configurations of points.

\subsection*{Acknowledgements}

The authors are indebted to Hiromu Tanaka for the very useful discussion and
informing them of the current state of the art of the base point free theorem on surfaces. They also thank Kazuhiro Konno for informing them of the works by Noboru Nakayama and Yoichi Miyaoka.
S.~O.~ was partially supported by Grants-in-Aid for Scientific Research
(16H05994,
16K13746,
16H02141,
16K13743,
16K13755,
16H06337)
and the Inamori Foundation.

\subsection*{Notation and conventions}
The ground field, which will be assumed to be
algebraically closed of arbitrary characteristic unless otherwise stated,
will be denoted by $\bfk$.
A \emph{curve} (respectively, \emph{surface}) is a projective and geometrically integral scheme over $\bfk$ of dimension one (resp. two). For a vector bundle($=$locally free sheaf) $E$ on a scheme $C$, the \emph{associated projective bundle} will be defined as
$
 \bP _{ C } \lb E \rb = \Proj _{ C } \lb \Sym E \rb
$.
An $\bR$-Cartier divisor $D$ on a projective scheme is said to be \emph{nef}
(respectively \emph{big}) if the intersection number with any integral curve is
non-negative (resp. if there is a positive constant $c \in \bR_{>0} $ such that
$
 \dim_ \bfk H^0(\lfloor k D \rfloor ) > c k ^2
$
holds for any sufficiently large integer $k$).
We will use the shorthand notation
$
 \hom =\dim \Hom, h ^{ i } = \dim H ^{ i },
$
etc.

\section{Stability of vector bundles on curves}

We prove some facts about stability of bundles on curves.
Throughout this section $C$ is a smooth projective curve, $E$ is a vector bundle of rank $r$ on it, and $\pi \colon X \coloneqq \bP _{ C } (E) \to C$ is the projective bundle associated to $E$.
\begin{definition}\label{Def:stab}
The \emph{slope} of $E$
is defined by
\begin{align}
 \mu \lb E \rb \coloneqq \frac{\deg E}{ \rank E}.
\end{align}
$E$ is said to be \emph{semi-stable}
(respectively, \emph{stable}) if for any subsheaf
$
 0 \subsetneq V \subsetneq E
$
the inequality
\begin{align*}
 \mu(V) \le \mu(E) \quad ( \mbox{resp.} \  < )
\end{align*}
holds. $E$ is \emph{strongly semi-stable} if the vector bundle
\begin{align*}
E^{(r)} \coloneqq (F^r)^* E =
(\overbrace{F\circ \dots \circ F} ^{r \ \mathrm{times}})^*E
\end{align*}
over $C$ is semi-stable for all $r\ge 0$, where
$
F \colon C \to C
$
is the Frobenius morphism.
\end{definition}
Note that if $\operatorname{char}(\bfk)=0$, there is no difference between semi-stability and strong semi-stability.

\begin{proposition}\label{pr:stable_property}
\
\begin{enumerate}
\item\label{it:symm_inherits_stability}
If $E$ is strongly semi-stable,
the $n$-th symmetric power $ S ^{ n } ( E ) $ is also strongly semi-stable.

\item
Assume $g(C) \le 1$. Then $E$ is semi-stable if and only if it is strongly semi-stable.

\end{enumerate}
\end{proposition}

\begin{proof}
(1) is the consequence of \cite[Theorem $7.2$]{MR1488349}.
See \cite[THEOREM 2.9]{MR2483939} or \cite[Theorem 2.1]{MR714755} for (2).
\end{proof}

\begin{proposition}\label{pr:hom}
Let $E$ and $F$ be stable vector bundles of the same slope.
Then every non-zero map $f \colon E \to F$ is an isomorphism and
$ \hom (E,F)\le 1$.
\end{proposition}

\begin{proof}
This is an easy consequence of \cite[Proposition 5.3.3]{MR1428426} and
\cite[Corollary 5.3.4]{MR1428426}.
\end{proof}

\begin{proposition}\label{pr:stability_g=2}
Suppose
$
 g ( C ) \ge 2
$
and
$
 H ^{ 0 } ( X, \cO _{ X } \lb - m K _{ X } \rb ) \neq 0
$
for some positive integer $m$. Then $E$ is not strongly semi-stable.
\end{proposition}

\begin{proof}
Recall that the canonical sheaf of $X$ admits the following isomorphism.
\begin{align}
 \cO _{ X } (K_X) \simeq
 \cO _{ X } ( - r ) \otimes \pi ^* ( \cO _{ C } ( K _{ C } ) \otimes \wedge ^{ r } E)
\end{align}
Hence a non-zero global section of
$
 \cO _{ X } \lb - m K _{ X } \rb
$
corresponds to a non-trivial morphism
\begin{align}
 \pi^*(\cO _{ C }(mK_C) \otimes (\wedge^r E)^{\otimes m}) \to \cO _{ X } (rm),
\end{align}
whose adjoint is a non-trivial morphism
\begin{align}
 \cO _{ C } ( m K _{ C } ) \otimes (\wedge^r E)^{\otimes m} \to \pi_*\cO _{ X } ( r m ) \simeq S ^{ r m } ( E ),
\end{align}
which necessarily is an injective morphism.
On the other hand, by \pref{lm:symmetric_product} below, the slopes of these two bundles are
$
 m ( 2 g - 2 + \deg E )
$
and
$
 m ( \deg E )
$, respectively. Hence $S^{rm}(E)$ is unstable. By \pref{pr:stable_property} (\pref{it:symm_inherits_stability}), we see that $E$ is not strongly semi-stable.
\end{proof}

\begin{lemma}\label{lm:symmetric_product}
The $n$-th symmetric power $ S ^{ n } ( E ) $ is a vector bundle of rank $ \binom{r-1+n}{n} $
and
$
\deg S^n (E) = \binom{r-1+n}{r} \deg E
$.
\end{lemma}
\begin{proof}
This is well known.
\end{proof}

We next discuss the case when $g ( C ) = 1$.

\begin{proposition}\label{pr:stability_g=1}
Suppose
$
 g \lb C \rb = 1
$.
If $-K_X$ is big, then $E$ is not strongly semi-stable.
\end{proposition}

\begin{proof}
It is sufficient to show that $S^{rm}(E)$ is unstable for some $m$ by \pref{pr:stable_property}.
We will assume that $S^{rm}(E)$ is semi-stable for any $m$ and show
that $- K _{ X }$ is not big. Note that there exists the isomorphism
\[
 \pi _{ * } \cO _{ X } ( - m K_X )
 \simeq
 S^{rm}(E) \otimes L,
\]
where $L = (\wedge ^r E)^{-m} $.
This implies that it is enough to show the convergence
\begin{align}
 \lim_{m \to \infty} \frac {h^0(C, S^{rm}(E) \otimes L) }{m^r} = 0.
\end{align}

Let
\begin{align}
 0 \subset F_1 \subset  F_2 \subset \dots \subset F_{\ell} = S^{rm}(E) \otimes L
\end{align}
be a Jordan-H\"older filtration of $S^{rm}(E) \otimes L$,
so that for each $i$ there is an exact sequence
\begin{align}\label{ex:T_i}
 0 \to F_{i-1} \to F_i \to  \gr _i \to 0
 \end{align}
such that $ \gr _i$ is stable of slope $\mu( \gr _i) = \mu (S^{rm}(E) \otimes L)=0$. Note that
$
 \ell \le \rank (S^{rm}(E) \otimes L) = \binom{r-1+mr}{mr}
$.

Let $\xi \in \Ext^1( \gr _i,F_{i-1})$ be the extension class of \eqref{ex:T_i}, and consider the affine line
\begin{align}
 B \coloneqq \bfk \xi \subset \Ext ^{ 1 } \lb gr _{ i }, F _{ i - 1 } \rb.
\end{align}
Then we can construct the tautological vector bundle $\cE$ on $C \times B$ satisfying
\[
\cE |_{C \times \{t\} } \simeq
\begin{cases}
 F_i & t \neq 0 \in B\\
 F_{i-1} \oplus  \gr _i & t = 0 \in B
\end{cases}\]
(see \cite[Section (7.3)]{MR1428426}).
Let $p_2$ be the natural projection from $C \times B$ to B.
Then by taking a general $t \in B$, we obtain the following inequality
\[
 h^0(C, F_i)
 = h^0 (f^{-1}(t), \cE | _{ C \times \lc t \rc })
 \le h^0 ( f^{-1}(0), \cE | _{ C \times \lc 0 \rc } )
 = h^0 (C, F_{i-1}\oplus  \gr _i)
\]
by the upper semicontinuity theorem
(see \cite[Section 5 COROLLARY 1]{MR2514037}).
This inequality holds for all $i$, and we have
\begin{align}\label{eq:jj}
 h^0(C, S^{rm}(E) \otimes L) \le
 h^0 \lb C, \bigoplus _{i=0} ^ \ell  \gr _i \rb
 = \sum _{i=0} ^ \ell h^0 (C,  \gr _i).
\end{align}
Since
$
 h^0 (C,  \gr _i) =\dim_\bfk \Hom (\cO _{ X },  \gr _i) \le 1
$ by \pref{pr:hom}, we obtain the inequality
\[
 \sum_{i=0} ^ \ell h^0 (C,  \gr _i) \le \binom{r-1+mr}{mr}.
\]
Thus we proved
$\displaystyle \lim_{m \to \infty} \frac {h^0(C, S^{rm}(E) \otimes L) }{m^r} = 0.$
\end{proof}

\begin{remark}
$E$ is in fact unstable because of \pref{pr:stable_property}.
\end{remark}

\begin{corollary}\label{cr:sss_implies_big}
$E$ is not strongly semi-stable if $-K_X$ is big.
\end{corollary}

\begin{proof}
This follows from \pref{pr:stability_g=2} and \pref{pr:stability_g=1}.
\end{proof}

\section{Geometrically ruled surface with big anti-canonical line bundle}

In the rest of this paper, let $C$ be a smooth projective curve of genus
$
 g \ge 1
$
and $E$ be a vector bundle on $C$ of rank $r$ unless otherwise stated. Let
$
 \pi \colon X = \bP _{ C } \lb E \rb \to C
$
be the projective bundle associated to $E$.
In this section we investigate the question when
$
 - K _{ X }
$
is a big line bundle.

First we consider vector bundles which are isomorphic to direct sums of line bundles.

\begin{proposition}\label{pr:AA}
Consider the direct sum
 $E = \bigoplus _{i=1} ^{r} L_i $
of line bundles $L_i$ on $C$ satisfying
$
 \deg L _{ r } \le
 \deg L _{ r - 1 } \le
 \cdots \le
 \deg L _{ 1 }.
$
Then $-K_X$ is big if
\begin{align}\label{eq:higher_rank_condition}
 ( r - 1 ) \deg L_1 - \sum _{ i = 2 } ^{ r } \deg L_i > 2 g - 2.
\end{align}
\end{proposition}

\begin{proof}
Note first that
\begin{align*}
\begin{split}
 h^0 (X, \cO_X(-mK_X)) = h^0 ( C, \pi_* \cO_X(-mK_X) ) \\
 = h^0 ( C , S^{rm} (E) \otimes (\wedge^r E \otimes \cO_C (K_C))^ {-m})
 = \sum _{k \in A_m}  h^0 (C, \cL_k ),
\end{split}
\end{align*}
where
\begin{align}
\begin{split}
 A_m \coloneqq \lc k = (k_1, \dots ,k_r) \in \bZ ^{r} _ {\ge 0} \mid \sum  k_i = rm \rc,\\
 \cL_k \coloneqq L_1 ^{(k_1-m)} \otimes \dots \otimes L_r ^{(k_r - m)} \otimes \cO_C (-mK_C).
\end{split}
\end{align}

Assume \pref{eq:higher_rank_condition}.
We may assume $\deg L_i \ge 0 $ without loss generality by replacing $L_i$ with $L_i \otimes L_r$.
Take a sufficiently small rational number
$
\delta >0
$
such that
\[
a \coloneqq  (1-\delta) (r-1) \deg L_1 - \sum _{i=2} ^{r} \deg L_i - (2g-2) > 0.
\]
If
$
0 \le k_i \le \delta m
$
for all
$
i \ge 2
$,
then
$
 (r - (r-1) \delta) m \le k_1 \le rm
$
and hence
\begin{align*}
h^0 (\cL_k) = \sum_{i=1} ^{r} (k_i - m) \deg L_i -m (2g-2)+ 1 - g + h^1 (\cL_k) \\
\ge ( (r - (r-1) \delta) m - m ) \deg L_1  -m \sum_{i \ge 2} \deg L_i -m (2g-2) -g \\
\ge m \lb (1-\delta) (r-1) \deg L_1 - \sum _{i=2} ^{r} \deg L_i - (2g-2) \rb - g \\
\ge am -g.
\end{align*}
Since
$
\# \lc k \in A_m \mid 0 \le k_i \le \delta m \quad \forall i \ge 2 \rc \ge (\delta m - 1) ^ {r-1}
$,
we deduce
\[
h^0 (X, \cO_X (-mK_X)) \ge (am-g) (\delta m-1) ^{r-1} = a s^{r-1} m^r + O (m^{r-1}).
\]
Hence $-K_X$ is big.
\end{proof}

We next show the converse of \pref{pr:AA}

\begin{proposition}\label{pr:BB}
Under the same assumption of \pref{pr:AA},
$-K_X$ is not big if
\begin{align}\label{eq:higher_rank_condition_2}
(r-1) \deg L_1 - \sum _{i=2} ^{r} \deg L_i \le 2g-2.
\end{align}
\end{proposition}

\begin{proof}
we may assume that
$
0 \le \deg L_1
$
and
$
\deg L_i \le 0
$
for all
$
i \ge 2
$
without loss of generality. Set
$
\cL_k \coloneqq L_1 ^{(k_1-m)} \otimes \dots \otimes L_r ^{(k_r - m)} \otimes \cO_C (-mK_C)
$
as in the proof of \pref{pr:AA}.

Note that if $h^0(\cL_k)>0$ then
\begin{align}\label{eq:dd}
 h^1(\cL_k) = h^0(\cO_C(K_C) \otimes \cL_k ^{ - 1 }) \le h^0(\cO_C(K_C)) = g.
\end{align}
Hence for all $k \in A_m$, either
$
 h ^{ 0 } \lb D _{ i } \rb = 0
$
or

\begin{align*}
h^0 (\cL_k) = \sum_{i=1} ^{r} (k_i - m) \deg L_i -m (2g-2)+ 1 - g + h^1 (\cL_k) \\
\le (k_1 -m) \deg L_1 + \sum_{i=2} ^{r} (k_i - m) \deg L_i -m (2g-2) + 1 + h^1 (\cL_k) \\
\le (rm -m) \deg L_1 - m \sum_{i=2} ^{r} \deg L_i -m (2g-2) + 1 + h^0 (K_X) \\
\le m \lb (r-1) \deg L_1 - \sum _{i=2} ^{r} \deg L_i - (2g-2) \rb + 1 + h^0 (K_X) \\
\le 1 + h^0 (K_X).
\end{align*}

Then $-K_X$ is not big because
$
 \# A_m \le (rm)^{r-1}
$.
\end{proof}

We obtain the following theorem by combining \pref{pr:AA} with \pref{pr:BB}.

\begin{corollary}\label{cr:main_theorem_split_case}
Under the assumptions of \pref{pr:AA}, $-K_X$ is big if and only if
\begin{align}\label{eq:higher_rank_condition 3}
 (r-1) \deg L_1 - \sum _{i=2} ^{r} \deg L_i > 2g-2.
\end{align}
\end{corollary}

Let $r =2$ and
consider an exact sequence of vector bundles on $C$ as follows,
where $L$ and $M$ are both line bundles.
\begin{align}\label{eq:extension}
0 \to L \to E \to M \to 0
\end{align}.
Below is the main theorem of this section.
\begin{theorem}\label{th:main_theorem}
Let
$
 E
$
be an unstable vector bundle of rank $2$ on $C$.
Then $-K_X$ is big if and only if there are line bundles
$L$ and $M$ on $C$ such that
$
 E \simeq L \oplus M
$
and
$
 \deg L- \deg M > 2 g - 2
$.
\end{theorem}

\begin{proof}
The 'if' direction is already shown in \pref{cr:main_theorem_split_case}.
To show the 'only if' direction, assume that $-K_X$ is big. We then obtain an exact sequence \eqref{eq:extension}
such that $\deg L > \deg M$ as the Harder-Narasimhan filtration of $E$.

Let $\xi \in \Ext^1(M,L)$ be the corresponding extension class and set
$
 B \coloneqq \bfk \xi \simeq \bA ^1
$.
Then we have a rank $2$ vector bundle $\cE$ on $C \times B$ such that
\begin{align}
\cE |_{C \times \{t\} } \simeq
\begin{cases}
 E & ( t \neq 0 )\\
 L \oplus M & ( t = 0 ).
\end{cases}
\end{align}
Consider the following diagram.
\begin{align}
 \xymatrix{
    \cX:=\bP_{C \times B}(\cE) \ar[r]^-{\pi} \ar[dr]_{f} & C \times B \ar[d]^{p_2} \\
    &  B
 }
\end{align}
Note that
$
 \cX _{ 0 } \simeq \bP_C( L \oplus M)
$
and
$
 \cX _{ t } \simeq \bP_C(E)\ (t \neq 0)
$.
Furthermore we have
\[
 \cO_\cX(K_{\cX})|_{ \cX _{ t } }=\cO_\cX(K_{ \cX _{ t } })\ \ (t \in B),
\]
where $\cO_\cX (K_{\cX})$ is the canonical sheaf of $\cX$.
By using the upper semicontinuity theorem \cite[2.5 COROLLARY 1]{MR2514037} we obtain the inequality
\begin{align*}
 h^0( \cX _{ t }, \cO_\cX( m K_{\cX})| _{ \cX _{ t } } )
 \le
 h^0 ( \cX _{ 0 }, \cO_\cX( m K_{\cX}) | _{ \cX _{ 0 } })
\end{align*}
for any $ m > 0 $ and general $t \in B$. Hence we obtain the inequality
\begin{align}
 h ^{ 0 } \lb X, \cO _{ X }(- m K _{ X }) \rb
 \le
 h ^{ 0 } \lb \bP _{ C } \lb L \oplus M \rb, \cO_{ \bP _{ C } \lb L \oplus M \rb }( - m K _{ \bP _{ C } \lb L \oplus M \rb }) \rb
\end{align}
for any
$
 m > 0
$, so that the bigness of $-K_{X}$ implies that of
$
 - K _{\bP_C (L \oplus M )}
$.
Thus we obtain the inequality
$
 \deg L - \deg M > 2 g - 2
$
from \pref{cr:main_theorem_split_case}.
This in fact implies that \eqref{eq:extension} is a trivial extension, since
\begin{align}
 \ext _{ C } ^{ 1 } \lb M, L \rb
 = h^1 \lb C, L \otimes M ^{ - 1 } \rb
 = h^0 \lb C, \cO _{ C } ( K _C ) \otimes L^{-1} \otimes M \rb = 0.
\end{align}
For the last equality, use
$
 \deg \lb \cO _{ C } ( K _C ) \otimes L^{-1} \otimes M \rb < 0
$.
Hence
$
 E \simeq L \oplus M
$,
concluding the proof.
\end{proof}

\begin{remark}\label{rm:nakayama's_results}
In fact \pref{th:main_theorem} and \pref{cr:sss_implies_big} are special cases of \cite[Chapter IV. 3.7. Lemma]{MR2104208}.
Let
$
\pi \colon X \coloneqq \bP _{ C } (E) \to C
$
be the projective bundle. Consider the Harder-Narasimhan filtration of $E$
\begin{align}
0 \subset E_1 \subset E_2 \subset \dots \subset E_k = E,
\end{align}
and assume that the successive quotients
$
 \gr _i \coloneqq E_i / E_{i-1}
$
are strongly semi-stable.
By combining \cite[Chapter IV. 3.7. Lemma]{MR2104208} with the well-known fact that the big cone is the interior of the pseudo-effective cone, we see that
the line bundle $\cO_{X} (m) \otimes \pi^* D$ on $X$,
where
$
m \in \bZ
$
and
$D$ is a line bundle on $C$, is big if and only if
\begin{align}\label{eq:big_slope}
 \deg D < m \mu ( \gr _1).
\end{align}

This implies that if $E$ is strongly semi-stable, $-K_X$ cannot be big as long as $g(C) \ge 1$.
Under the assumption of \pref{th:main_theorem}, it follows that
$-K_X$ is big
if and only if
$
 \deg L - \deg M > 2 g - 2
$,
since $L =  \gr _1$ and $M =  \gr _2$.

The proof of \cite[Chapter IV. 3.7. Lemma]{MR2104208}, in turn, is based on a numerical criterion for semi-stability \cite[Theorem 3.1]{MR946247}. Compared to this, our arguments above are more elementary and naive.
\end{remark}

\begin{remark}
If the rank of an unstable vector bundle $E$ is at least three,
the bigness of $-K_X$ does not necessarily imply that
$E$ is isomorphic to a direct sum of semi-stable vector bundles.
To give such an example, let $g \ge 2$,
$L$ be a line bundle of degree $g$ on $C$ such that $h^0 (C, \omega_C \otimes L^{-1}) >0 $,
and
$E' \coloneqq \cO_C \oplus \cO_C $.
Note that $E'$ is semi-stable.
Let $E$ be the extension of
$E'$ by $L$ given by a non-trivial element of
\begin{align*}
\Ext ^1 (E', L)
\simeq  H^1 (C, ( E ' ) ^{ - 1 } \otimes L) \\
\simeq H^0 (C, \omega_C \otimes ( ( E ' ) ^{ - 1 } \otimes L)^{-1}) ^{ \vee }\\
\simeq H^0 (C, \omega_C \otimes ((\cO_C \oplus \cO_C) \otimes L)^{-1} ) ^{ \vee } \\
\simeq H^0 (C, \omega_C \otimes L^{-1}) ^{ \vee } \oplus H^0 (C, \omega_C \otimes L^{-1}) ^{ \vee } \neq 0.
\end{align*}
By the same arguments in \pref{rm:nakayama's_results}, we see that $ -K_{\bP_C(E)} $ is big. To see that $E$ does not admit a decomposition into semi-stable vector bundles, one can use the uniqueness of the Harder-Narasimhan filtration.
Note that if $g = 1$, there are no such examples (see \cite[Theorem 10]{MR2264108}).
\end{remark}

\begin{corollary}\label{cr:structure_of_Frobenius_pullback}
Let
$
 E
$
be a vector bundle of rank $2$ on $C$.
If
$
 - K _{ X }
$
is big, there exists an integer
$
 e \ge 0
$
such that the Frobenius pull-back
$
 \lb F ^{ e } \rb ^{ * } E
$
is isomorphic to a direct sum of line bundles
$
 L ' \oplus M '
$
such that
$
 \deg L ' - \deg M ' > 2 g - 2
$.
\end{corollary}

\begin{proof}
By \pref{cr:sss_implies_big}, we see that $E$ is not strongly
semi-stable. Hence there exists a positive integer
$
 e \ge 1
$
such that the Frobenius pull-back
$
 E ' \coloneqq \lb F ^{ e } \rb ^{ * } E
$
is unstable. Set
$
 f \coloneqq F ^{ e }
$,
$
 Y ' \coloneqq \bP_C ( E ' )
$, and name the morphisms as in \pref{fg:first_diagram} below.

By applying the exact functor
$- \otimes _{ C } \pi_* \cO_Y (-m K_Y)$
to the injective homomorphism
$
 \cO_C \hookrightarrow f_* \cO_C
$
and then taking $H ^{ 0 }$, we obtain the inequality
\begin{align}
 h^0(\pi_* \cO_Y (-m K_Y)) \le h^0(f^*( \pi_* \cO_Y (-m K_Y))).
\end{align}
On the other hand, for each integer $m > 0$ we have
\begin{align}
 \pi'_* ( \cO_{Y'}(- m K _{ Y ' }) ) =  f^*( \pi_*\cO_{Y'} (-m K_Y)) \otimes
 (f^* \omega_C \otimes \omega_C^{-1}) ^{ \otimes m }.
\end{align}
Since
$
 \deg (f^* \omega_C \otimes \omega_C^{-1}) > 0
$,
we have
$
 H ^{ 0 } \lb \lb f^* \omega_C \otimes \omega_C^{-1} \rb ^{ \otimes m }\rb
 \neq 0
$
for any sufficiently large $m$. By the similar arguments as above,
we obtain the following inequality.
\begin{align}
 h^0(f^*( \pi_* \cO_Y (-m K_Y))) \le h^0( \cO_{Y'}(- m K _{ Y ' } ))
\end{align}
Thus we see that $- K _{ Y ' }$ is big, and the claim now follows from \pref{th:main_theorem}.
\end{proof}

\section{Numerically trivial line bundles on normal surface with nef and big anti-canonical bundle}
\label{sc:Numerically trivial line bundles on normal surface with nef and big anti-canonical bundle}

In this section we prove that a numerically trivial line bundle on a
normal projective surface with nef and big anti-canonical divisor
is trivial. This is a result which is proved for some partial cases in
the preceding works \cite{MR3319838, MR3383599}.
In fact we give two different proofs.
In the first subsection, we give a proof as an application of our results
obtained so far. The point is that the minimal model
(in the classical sense) of the minimal resolution has big anti-canonical
line bundle, so that we can apply our results to it.
In the second subsection, we give a proof by combining known results.

\subsection{Proof via \pref{cr:structure_of_Frobenius_pullback}}

\begin{theorem}\label{th:adjunction}
Let $X$ be a normal projective surface
such that $-(K_X+\Delta)$ is a nef and big \bR-Cartier divisor,
where $\Delta$ is an \bR-divisor satisfying $\lfloor \Delta \rfloor = 0$.
Then there exists a positive integer $m > 0$ which depends only on $X$ such that
any numerically trivial line bundle $\cL$ on $X$ satisfies
$
 \cL ^{ \otimes m } \simeq \cO _{ X }
$.
\end{theorem}

\begin{proof}
If
$
 X
$
is rational, the assertion is obvious. Suppose otherwise.
Let $\varphi \colon {\Xtilde} \to X$ be the minimal resolution of $X$ and set
$\Delta' \coloneqq \varphi^{-1}_* \Delta$.
Then we have the canonical bundle formula as follows.
\begin{align}\label{eq:22}
 K_{\Xtilde}+\Delta' \equiv \varphi^* (K_X + \Delta) + \sum a_i E_i
\end{align}
Since $ a_i \le 0$ (see \cite[Corollary $4.3$]{MR1658959}) and
$- ( K_X + \Delta )$ is big, so is $-K_{\Xtilde}$.

\begin{figure}
\begin{align}
 \xymatrix{
   & {\Xtilde} \ar[r]^{\varphi} \ar[d]^{\varepsilon} &  X\\
 Y' \ar[r]^{f'}\ar[d]^{\pi'}  &  Y\ar[d]^{\pi} \\
 C\ar[r]^{f}  &  C
  }
\end{align}
\caption{\label{fg:first_diagram}
\end{figure}

\begin{figure}
\begin{align}
 \xymatrix{
 & {\Xtilde}\ar[r]^{\varphi} & X\\
(\Ctilde)_{\operatorname{norm}}\ar[r]^{\psi}
& \Ctilde\ar[r]^{\varphi} \ar@{^{(}->}[u]\ar[d]^{\varepsilon}  &\{pt\} \ar@{^{(}->}[u]  \\
s(C)\ar[u]^{\alpha} \ar@<0.5ex>[d]^-{\pi'} \ar[r]^{f'} &
\Cbar \ar[d]^{\pi}\\
C \ar@<0.5ex>[u]^-s  \ar[r]^{f} & C
}
\end{align}
\caption{\label{fg:main_diagram}
\end{figure}

Let
$
 \varepsilon \colon \Xtilde \to Y
$
be a composition of contractions of $(- 1)$-curves such that
$Y$ is a geometrically ruled surface $\bP_C(E)$ with
$
 g ( C ) \ge 1
$
(recall that $X$ is assumed to be non-rational).
Consider the canonical bundle formula
\begin{align}\label{eq:11}
K _{ \Xtilde } = \varepsilon^*  K_Y + F
\end{align}
with respect to $\varepsilon$, where $F$ is an effective
$\varepsilon$-exceptional divisor on $\Xtilde$.
Since $-K_{\Xtilde}$ is big, so is $-K_Y$.
By \pref{cr:main_theorem_split_case}, the pull-back
$
 E ' \coloneqq f ^{ * } E
$,
where
$
 f = F ^{ e }
$
is some iteration of the Frobenius morphism, is isomorphic to
a direct sum of line bundles
$
 L ' \oplus M '
$
such that
$
 \deg L ' - \deg M ' > 2 g - 2
$.

Let $s$ be the section of $\pi'$ corresponding to the natural projection $E' \to M'$.
Let
$
 \Cbar \subset Y
$
be the closed subscheme
$
 f ( s ( C ) ) \subset Y
$
equipped with the reduced structure, and
$\Ctilde$ be the strict transform of $\Cbar$ in ${\Xtilde}$.
These varieties and morphisms form the diagram
\pref{fg:main_diagram}.

\begin{claim}\label{cl:positivity}
$(\Ctilde. K_{\Xtilde}+\Delta ') > 0$.
\end{claim}
\begin{proof}
It is easy to see that
$
 f' _*[s(C)] = \lb \deg f'| _s(C) \rb [\Cbar]
$
and
$(\cO _{ Y ' }(1).s(C)) = \deg M'$. Hence
\begin{align}
\begin{aligned}
 \lb \deg f' | _C \rb (\Cbar. K_Y) = s(C).f'^*K_{Y} \\
= s(C).f'^* \cO _{ Y } (-2) + s(C). f'^* \pi^* (\wedge ^2 E \otimes\ \omega_{C}) \\
= s(C).\cO _{ Y ' }(-2) + s(C).\pi'^* (\wedge ^2 E' \otimes f^* \omega_{C})\\
= \deg L' - \deg M' + \lb \deg f \rb (2g-2) > 0,
\end{aligned}
\end{align}
so that
$
 (\Cbar. K_Y) > 0
$.
If $\Ctilde ^2\ge 0$, it follows from this inequality and \pref{eq:11}
that
$(\Ctilde. K_{\Xtilde}+\Delta') > 0$.
Now suppose $\Ctilde ^2<0$. Write
\[
 \Delta' = \alpha \Ctilde + R,
\]
where $0 \le \alpha <1$.
Then since $g(\Ctilde) \ge 1 $ and $\Ctilde ^2 < 0$, we have
\begin{align}
\begin{aligned}\label{neq:intersection}
(\Ctilde. K_{\Xtilde}+\Delta) = (\Ctilde^2) + (\Ctilde.K_{\Xtilde}) + (\alpha - 1)(\Ctilde ^2) + (\Ctilde.R) \\
=2g(\Ctilde)-2 + (\alpha-1)(\Ctilde^2) + (\Ctilde.R) >0.
\end{aligned}
\end{align}
\end{proof}

Consider the following equality obtained from \pref{eq:22}.
\[
(\Ctilde.K_{\Xtilde}+\Delta') = (\varphi_*(\Ctilde).K_X+\Delta) + \sum a_i(E_i. \Ctilde )
\]
Since $- (K _{ X }+\Delta)$ is nef, $(\varphi_*(\Ctilde).K_X+\Delta) \le 0$.
If $\Ctilde$ is not contracted by $\varphi$, then the second term of the right hand side is $\le 0$ since $a_i \le 0$ and $\Ctilde \neq E _{ i }$ for all $i$.
This contradicts \pref{cl:positivity}.
Hence $\Ctilde$ must be contracted to a point by $\varphi$.

Now we prove the theorem. Let $\cL$ be a numerically trivial line bundle on $X$.
There exists a numerically trivial line bundle $\cL_Y$ on $Y$ such that
$
 f ^* \cL = \varepsilon ^* \cL_Y
$.
Since
$
 \Pic  \bP _{ C } (E)  = \pi ^{ * } \Pic C \perp \bZ \cO ( 1 )
$,
there exists a numerically trivial line bundle
$\cL_C$ on $C$ such that $\cL_Y = \pi ^* \cL_C.$
It follows from \pref{fg:main_diagram} that
$
 (\varepsilon ^* \pi^* \cL_C) |_{\Ctilde} = \varphi^* \cL|_{\Ctilde} = 0
$.
Now let \[\psi: (\Ctilde)_{\operatorname{norm}} \to \Ctilde \]
be the normalization of $\Ctilde$.
Note that \[\varepsilon \circ \psi :(\Ctilde)_{\operatorname{norm}} \to \Ctilde \to \Cbar \]
is a normalization of $\Cbar$ as well.
These morphisms form the diagram \pref{fg:main_diagram}, where
$\alpha$ is the natural morphism induced by the universal property of the normalization. Then we have
\begin{align}
 \cO _{ C } \simeq s^* \alpha^* \psi^* ((\varepsilon ^* \pi^*) \cL_C|_{\Ctilde})
 \simeq s^* f'^* \pi^* \cL_C
 \simeq f^* \cL_C \simeq \cL_C ^{ \otimes \deg f }.
\end{align}
Therefore we can take
$
 m \coloneqq \deg f
$
when
$
 \operatorname{char} \lb \bfk \rb > 0
$,
and
$
 m \coloneqq 1
$
otherwise.
\end{proof}

We will use \cite[Lemma $2.1$]{MR1821186} in the proof of \pref{th:adj} below.

\begin{lemma}($=$\cite[Lemma 2.1]{MR1821186})\label{lm:freeness}
Let $X$ be a normal surface with irregularity
$
q(X) \coloneqq \dim \Pic^0 (X) = 0
$.
If no multiple $nK_Z$ with $n>0$ is effective, then $\Pic(Z)$ is a free abelian group of finite rank.
\end{lemma}

\begin{proof}
See \cite[Lemma 2.1]{MR1821186}.
\end{proof}

\begin{corollary}\label{cr:adj_closed}
Under the same assumption of \pref{th:adjunction}, $\Pic(X)$ is a free abelian group of finite rank.
\end{corollary}

\begin{proof}
If $X$ is rational, the assertion is obvious. Suppose otherwise.
It is enough to show that $q(X)=0$ by \pref{lm:freeness} and \pref{lm:non_effective}.
Note that in this case
$\Pic^0_X $ is smooth
since
$
 h ^2 ( \cO _{ X } )= 0
$
by \cite[PROPOSITION 9.5.19]{MR2222646}.

Moreover $\Pic^0_X$ is an abelian variety,
because $\Pic^0_X$ is projective by \cite[THEOREM 9.5.4]{MR2222646}.
We can now apply \pref{th:adjunction} to $(X, \Delta)$
to obtain an integer $m>0$ such that
$
L^{\otimes m} \simeq \cO_{X}
$
for all $L \in \Pic^0_X$.
Hence the $m$-th power map
\[
 \m \colon \Pic^0_X \to \Pic^0_X
\]
of the abelian variety $\Pic^0_X$ factors through the constant map to the identity.
On the other hand $\m$ is surjective and finite by generalities of abelian varieties; for example, it follows from \cite[p.59, Corollary 3]{MR2514037}
that the pull-back of an ample line bundle by $\m$ is ample.
Hence we see $q(X)=0$, concluding the proof.
\end{proof}

\begin{lemma}\label{lm:non_effective}
Suppose $-(K_X + \Delta)$ is big for some effective $\bR$-divisor $\Delta$.
Then $nK_X$ is not effective for all $n>0$.
\end{lemma}

\begin{proof}
Suppose
$
 n K _{ X } \sim D \ge 0
$
for a contradiction.
Since
$
 - \lb K _{ X } + \Delta \rb
$
is big,
$
 - n (K_X + \Delta) \sim _{ \bR } D ' \ge 0
$
for some integer $n > 0$ and an effective $\bR$-divisor
$
 D ' \neq 0
$.
Summing up, we see that the non-zero effective $\bR$-divisor
$
 D + D ' + n \Delta
$
is a principal $\bR$-divisor.
This contradicts the fact that an effective principal $\bR$-divisor
on a $\bfk$-scheme $X$ such that
$
 H ^{ 0 } \lb X, \cO _{ X } \rb = \bfk
$
is never effective unless it is trivial.
\end{proof}

We now generalize \pref{cr:adj_closed} to arbitrary fields.
In the rest of this section, let
$
\bfk
$
be not necessarily algebraically closed and
$
 X
$
be a projective, geometrically integral, and geometrically normal surface
over $\bfk$.
Let $\overline{\bfk}$ be an algebraic closure of $\bfk$, and
\begin{align}
 \pi \colon \Xbar = X \otimes _\bfk \overline{\bfk} \to X
\end{align}
be the natural projection. For a coherent sheaf
$
 \cF \in \coh X
$, we use the shorthand notation
$
 \overline{\cF} \coloneqq \pi ^{ * } \cF
$. Then by the standard descent theory we have a canonical isomorphism
\begin{align}\label{eq:ext}
 H^i(X, \cF)\otimes_\bfk \overline{\bfk}
 \simeq
 H^i(\Xbar, \overline{\cF} ).
\end{align}
In particular, $X$ has at worst rational singularities if and only if
$\Xbar$ does. Furthermore, note that if $L$ is a nef and big line bundle,
then so is $\pi^*L$.

\begin{lemma}\label{lm:field_ext}
Let
$\bfk$ and $X$ be as above.
Let
$
\Delta
$
be as in \pref{th:adjunction} and assume that
any irreducible component of $\Delta$ is geometrically reduced.
Then the \bR-divisor $\Deltabar = \pi ^{ * } \Delta$ on $\Xbar$
also satisfies
$\lfloor \Deltabar \rfloor = 0$
and
$-(K_\Xbar + \Deltabar)$ is nef and big.
\end{lemma}

\begin{proof}
Assume that $\Delta = \sum a_i D_i$, where $D_i$ is a prime divisor on $X$.
Let $\Deltabar$ = $\sum a_i \Dbar _{ i }$ where
$\Dbar _{ i } = D_i \otimes _\bfk \overline{\bfk} \subset \Xbar$. Since each $D_i$ is geometrically reduced and $\lfloor \Delta \rfloor = 0$, we have $\lfloor \Deltabar \rfloor = 0$. Then we only have to show that $-\pi^*(K_X + \Delta) = -(K_\Xbar + \Deltabar)$, but we can check this by restricting everything over the smooth locus of $X$.
\end{proof}

Now we are ready to prove \pref{th:adj}.

\begin{corollary}\label{cr:adj}
Under the same assumption of \pref{lm:field_ext}, $\Pic(X)$ is a free abelian group of finite rank.
\end{corollary}

\begin{proof}
Note that the natural homomorphism $\pi^* \colon \Pic (X) \to \Pic (\Xbar)$ is injective.
In fact, if $\pi ^*L = \cO_\Zbar$,
then $h^0(L) = h^0(L^{-1}) = 1$ and this implies that $L = \cO_Z$ since $H^0(Z, \cO_Z) = \bfk$.
Hence we may assume that
$
 \bfk = \bar{\bfk}
$ by \pref{lm:field_ext}, and this is done in \pref{cr:adj_closed}
\end{proof}

\subsection{Proof via combination of known results}
\label{sc:combination}

We show the \pref{cr:adj} by combining the following
two known results.

\begin{proposition}\label{pr:rational}
Under the same assumption of \pref{lm:field_ext},
we also assume that $X$ has at worst rational singularities.
Then $\Xbar$ is a rational surface.
In particular, $\Pic(X)$ is a free abelian group of finite rank.
\end{proposition}

\begin{proof}
Apply
\cite[Theorem $3.5$]{MR3383599} and \cite[Corollary 3.6]{MR3383599} to
$(\Xbar, \Deltabar)$.
Then the injectivity of $\pi^*$ in the proof of \pref{cr:adj} implies the second part.
\end{proof}

\begin{proposition}\label{pr:non_rational}
Let
$\bfk$ and $X$ be as \pref{lm:field_ext}.
Assume $X$ has non-rational singularities and
$nK_X$ is not effective for all $n>0$.
Then $\Pic(X)$ is a free abelian group of finite rank.
In particular if $-(K_X + \Delta)$ is big for some effective $\bR$-divisor $\Delta$
and $X$ has non-rational singularities,
$\Pic(X)$ is a free abelian group of finite rank.
\end{proposition}

\begin{proof}
The first part follows from \cite[Theorem 2.2]{MR1821186} and the injectivity of $\pi^*$
since
$
K_\Xbar \simeq \pi^ *K_X
$.
Then the second part follows from \pref{lm:non_effective}.
\end{proof}

\subsection{What if $\lfloor \Delta \rfloor \in \lc 0, 1 \rc$?}
\label{sc:Delta=1}

Consider the case when some of the coefficients of $\Delta$ is $1$; namely,
replace the assumption
$
 \lfloor \Delta \rfloor = 0
$
of \pref{cr:adj}
with the weaker condition
$
 \lfloor \Delta \rfloor \subset \lc 0, 1 \rc
$.
Even in this generality, if $-(K_X + \Delta)$ is ample, then \pref{th:adjunction} and \pref{cr:adj} are still true. The proof of \pref{th:adjunction} given above works in this generality without change.
On the other hand, they do not hold true if $-(K_X + \Delta)$ is only nef and big. Such an example is given in \pref{eg:counter} below.

In fact \pref{eg:counter} is typical in the following sense. Assume that $\Pic (X)$ is not a free abelian group of finite rank. Then by closely examining the proof of \pref{th:adjunction},
especially the computation \eqref{neq:intersection},
we can check that the following properties have to be satisfied.
\begin{itemize}
\item
$g(C)=1$.

\item
$\alpha=1$.

\item
$
 \Ctilde \cap \Supp R = \emptyset
$.
\end{itemize}
Furthermore it follows from \pref{pr:non_rational} that $X$ has at worst rational singularities. It would be interesting to classify these exceptional cases.

\begin{example}\label{eg:counter}
Let $X$ be the projective cone over a smooth plane cubic curve $C$ defined by an equation
$
 F \lb x, y, z \rb \in \bfk [ x, y, z ]
$.
Since $X$ is the singular hypersurface of
$
 \bP ^{ 3 } _{ x, y, z, w }
$
defined by $F$, we obtain
$
 -K_X = \cO_X( 1 )
$
by the adjunction formula \cite[Chapter II, Theorem 7.11]{MR0463157}.
As is well known, there is a birational contraction
\begin{align}
 \varepsilon \colon \Xtilde \coloneqq \bP _{ C } \lb \cO _{ C } \oplus \cO _{ C } ( 1 ) \rb\to X
\end{align}
which contracts the section
$
 E \subset \Xtilde
$
of $\pi$ corresponding to the quotient map
$
 \cO _{ C } \oplus \cO _{ C } ( 1 ) \to  \cO _{ C }
$
to the vertex of $X$.

By using the structure morphism
$
 \pi \colon \Xtilde \to C
$, we obtain
\begin{align*}
 \bR \Gamma \lb \Xtilde, \cO _{ \Xtilde } \rb \simeq \bR \Gamma \lb C, \bR \pi _{ * } \cO _{ \Xtilde } \rb
 \simeq
 \bR \Gamma \lb C, \cO _{ C } \rb \simeq \bfk \oplus \bfk [ - 1 ].
\end{align*}
Hence the Picard scheme of $\Xtilde$ is smooth of dimension one, so that $X$ admits a numerically trivial but non-trivial line bundle.

On the other hand, by applying the adjunction formula to the embedding
$
 E \hookrightarrow \Xtilde
$,
one can easily verify the equality
$
 - ( K _{ \Xtilde } +E ) = - \varepsilon ^{ * } K _X
$.
Note that the right hand side is nef and big, since
$
 - K _{ X } = \cO _{ X }( 1 )
$
is (very) ample. Hence the pair $(\Xtilde, E)$ is an example which does not satisfy the assertion of \pref{th:adjunction}.
\end{example}

\section{Some examples of ruled surfaces whose anti-canonical sheaf is big}
In this section we construct some examples of (not necessarily geometrically) ruled surfaces whose anti-canonical sheaf is big.
Note that such a surface is obtained from a geometrically ruled surface
(i.e. a minimal model in the classical sense) with big anti-canonical bundle by repeatedly blowing up smooth points. The examples below indicate that there are quite a few examples of such surfaces. This is comparable to the case of big rational surfaces \cite{MR2824848}.

We first give a bigness criterion for line bundles on geometrically ruled surfaces which is associated to decomposable rank two vector bundles.

\begin{proposition}\label{pr:crit_big}
Let
$
 L
$
be a line bundle on
$
 C
$
with
$
\deg L \ge 0
$.
Consider
$
 E \coloneqq L \oplus \cO _{ C }
$
and let
$
 \pi \colon X \coloneqq \bP_C(E) \to C
$
be the canonical projection. For a line bundle
$
 L _{ C }
$
on
$
 C
$,
the line bundle
$
 \cO _{ X } (n) \otimes  \pi^* L _{ C }
$
on
$
 X
$
is big if and only if $ n > 0 $ and
$
 n \deg L + \deg L _{ C } > 0
$.
\end{proposition}

\begin{proof}
This is proved by almost the same argument in the proof of \pref{cr:main_theorem_split_case} or
\pref{rm:nakayama's_results}. As we discussed in \pref{rm:nakayama's_results}, this is a special case of \cite[Chapter IV. 3.7. Lemma]{MR2104208}.
\end{proof}

Below is an immediate corollary of \pref{pr:crit_big}.

\begin{corollary}\label{cr:big}
Let
$
L, X
$
be as in \pref{pr:crit_big}. For a line bundle
$
 M
$
on
$
 X
$,
the line bundle
$
 \cO _{ X } (K_X) \otimes M^{-1}
$
is big if and only if either
$
 M \simeq \pi ^{ * } L _{ C }
$
for some line bundle
$
 L _{ C }
$
on
$
 C
$
such that
\begin{align}
 \deg L _{ C } + \deg L > 2 g - 2,
\end{align}
or
$
 M \simeq \cO _{ X } (1) \otimes \pi ^* L _{ C }
$
for some line bundle
$
 L _{ C }
$
on
$
 C
$
such that
\begin{align}
 \deg L _{ C } < - ( 2 g - 2 ).
\end{align}
\end{corollary}

Let $f \colon \Xtilde \to X $ be the blow-up of X at a finite set of points which is contained in the support of an effective divisor $D$.
If $-K_X - D$ is big, $-K_{\Xtilde}$ is also big.
In fact, the second term of the right hand side of the following equation is effective, and the first term is big.
\begin{align*}
-K_{\Xtilde} = - f^* K_X - E
= - f^* \lb K_X + D \rb + \lb  f^*D -E \rb.
\end {align*}
Furthermore, by the same argument, we can see that the bigness of the anti-canonical divisor is preserved under the successive blow-ups centered at points in the strict transforms of $D$.
Thanks to this observation, we obtain the following examples.

\begin{example}
Let
$
L, X
$
be as in \pref{pr:crit_big}, and assume moreover
$
\deg L > 2 g- 1
$.
Set
$
 k \coloneqq \deg L - (2g-1)
$
and choose distinct fibers
$
 F_1, \dots, F_k
$
of the morphism $\pi$. Let
$
 S \subset \bigcup _{ k } F_k
$
be a finite subset and let
$
 f \colon \Xtilde \to X
$ be the blow-up of $X$ along $S$.
Then
$
 - K _{ \Xtilde }
$
is big.
This follows from the bigness of $-K_X - \sum_i F_i $ and \pref{cr:big}.
Moreover, the anti-canonical divisor is still big whenever we blow up at points in the strict transform of $\bigcup _{ k } F_k $.
\end{example}

\begin{example}
Suppose $g(C)=1$ and $\deg L = 1$.
Let $D$ be an effective divisor corresponding to a global section of
$
 \cO _{ X }(1) \otimes \pi ^* L ^{ - 1 }
$
(consider the case when it exists).
Then $-K_X - D$ is big by \pref{cr:big}, so that for the blow-up $\Xtilde$ of $X$ in a finite set of points on $D$
the anti-canonical sheaf $-K_{\Xtilde}$ is always big.
\end{example}

\bibliographystyle{amsalpha}

redpajama_set_name: RedPajamaArXiv


text:
Виктор Горидсен Фишер (роден на 9 юни 1994 в Орхус, Дания) е датски футболист, играе като нападател и се състезава за английския Мидълзбро.

Клубна кариера

Ранна кариера
Фишер започва да тренира футбол в скромния отбор на Лисенг ИФ. След това се премества в школата на Орхус, където прекарва пет години, преди да премине в школата на друг датски тим – Митюлан.

Аякс
Фишер преминава в школата на холандския гранд Аякс, където заиграва за А1 селекцията на "амстердамци". Договорът му е за три сезона, като изтича в края на юни 2014 г.

Фишер прави дебюта си за Аякс в предсезонната подготовка преди началото на сезон 2012/13 срещу СФ Хойзен и вкарва гол в 63-тата минута, а отбора му побеждава с 3 – 1. Дебютът си в Ередивиси прави на 20 октомври 2012 г., влизайки в 84-тата минута на мача срещу Хераклес. Мачът завършва 3 – 3. На 31 октомври 2012 г. изиграва първия си мач като титуляр в официален мач и също така отбелязва първия си гол за Аякс в официален мач при победата с 2 – 0 над ОНС Снеек за Купата на Холандия. Дебютът си в Шампионска лига прави при гостуването в Англия на Манчестър Сити. Появява се на терена в 87-а минута на мястото на сънародника си Кристиан Поулсен, а мача завършва 2 – 2. Първият си мач като титуляр в Ередивиси записва на 11 ноември 2012 г. срещу ПЕК Зволе. Аякс побеждава с 2-4 като гост, а Фишер реализира два гола в мача.

Статистика

 Статиктиките са валидни до 15 ноември 2012 г.

Национален отбор
Фишер е представлявал Дания до 16, до 17 и до 19 години. За отбора на Дания до 17 години има отбелязани 20 гола в 30 срещи. Част е от отбора на Дания до 17 години, който стига до полуфиналите на Европейското първенство до 17 години през 2011 г. През 2012 г. дебютира за отбора на Дания до 21 години.

Бележки

Външни препратки
 Профил в transfermarkt.co.uk

Датски футболисти
Футболисти на АФК Аякс
Футболисти на ФК Мидълзбро
Родени в Орхус

redpajama_set_name: RedPajamaWikipedia


The UK economy is on the cusp of a new digital revolution which will see a substantial amount of personal data collected, stored and processed. Data protection laws were originally created to increase digital innovation in businesses whilst ensuring consumer privacy and protecting data from cyber-attacks - making Britain a safer place online.
It has been a generation since the UK data laws were updated and yesterday, the government released a new Data Protection Bill outlining protections for the country's research, financial services, journalism and legal services. As part of the bill coming into force next May, EU's General Data Protection Regulation legislation will be maintained after Brexit, therefore now would be a good time for businesses to prepare for GDPR compliance.
Under the new law, the government is empowering people to take control with new rights to access, move and delete their personal data with the 'right to be forgotten'. However to protect data processing that is "vital" to our economy, the law does contain exemptions negotiated from the GDPR. Exemptions that previous have worked well in the Data Protection Act will be preserved to continue supporting the UK's leading services.
Culture Secretary, Karen Bradley said: "The Data Protection Bill will give people more control over their data, support businesses in their use of data, and prepare Britain for Brexit.
Perhaps the most significant change - businesses that suffer a data breach could be fined up to £17m or 4% of global turnover. In comparison, the current maximum fine for breaking data protection law is £500,000.
For a further overview of the new Data Protection Bill, you can read this fact sheet.


THE PERFECT HOME...Country riverfront estate offering unparalleled craftsmanship and amenities. Nestled within 6+ acres of peace, quiet & seclusion with 600' +/- on the Caloosahatchee River, part of the intracoastal waterway. The custom 3 story 7,000+ SF, 4BR/4BA home, completely remodeled in 2011 and exterior in 2018, boasts of versatile interior space, including multiple living areas, formal & informal dining areas, gourmet kitchen, fireplace, wet bar, media room, 2 master bedrooms and more! The home is attached by breezeway to a 3 vehicle over-sized garage with Tesla charging station and an upstairs finished bonus room with Italian tile flooring and air conditioning for exercise, office, or potential guest quarters. Exterior of home has Premier Vue PGT impact resistant windows and was completely renovated this year with cement board siding and engineered composite decking on front porch and balcony. Amenities include 50' lap pool; covered dock with composite decking, water, electric & 18K/lb lift & jet ski lift; barn; gazebo; greenhouse; and tropical landscaping. Site will allow for 2 add'l homes. This is for active & multi-faceted people who enjoy pursuing multiple interests!


text:
Home » Italy Hotels » The 9 Best Places to Stay Nearest to Riomaggiore (Cinque Terre) Train Station
The 9 Best Places to Stay Nearest to Riomaggiore (Cinque Terre) Train Station
The best guesthouses and apartments within 200 meters (3 minutes' walk) of the railway station in Riomaggiore (Cinque Terre).
There are a number of accommodations with great ratings and reviews within 200 meters, or around 3 minutes' walk, of the train station in Riomaggiore, the southernmost village of the Cinque Terre. The guesthouses and apartments shortlisted below all have a rating of at least 8.5/10 at Booking.com and at least 4/5 in TripAdvisor and Google Reviews. The best three are described in more detail below and then all nine are summarized in the table that follows.
1. Villa Pecunia
This guesthouse on Via Telemaco Signorini, 380 offers double rooms. It is not only a short walk from the train station; it's also close to the center of town. All rooms have private entrance and a balcony — perfect for watching the sunset — as well as a mini fridge, A/C and Wifi. Children and pets are not allowed. Depending on the season and occasion, a one-night stay at this hotel starts at €250 – €450.
80 meters from Riomaggiore (Cinque Terre) train station
2. La Casa di Luna
This guesthouse on Via Telemaco Signorini, 527 offers double rooms with A/C, private entrances, a small garden and a small kitchen where guests can prepare meals. It is only a 5-minute walk to the beach. Children are welcome. Pets are allowed, subject to possible extra charges. Depending on the season and occasion, a one-night stay at this hotel starts at €100 – €140.
120 meters from Riomaggiore (Cinque Terre) train station
9 / 10 at Booking.com, 4.8 / 5 at Google Reviews
3. The 5 Terre
This guesthouse on Via Telemaco Signorini, 506 offers double rooms with private entrance and A/C. Children and pets are not allowed. Depending on the season and occasion, a one-night stay at this hotel starts at €75 – €113.
9 / 10 at Booking.com, 4.5 / 5 at TripAdvisor, 5 / 5 at Google Reviews
Best places to stay within 200 meters of Riomaggiore train station
L'oblò
Dreaming Rooms
Sofia Holiday
Camere e Appartamenti Riomaggiore
Jolly Roger Riomaggiore
Villa Pecunia
Sollevante Guest House
La Casa di Luna
The 5 Terre

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: Telerik MVC Disable DatePicker using javascript I need to disable Telerik MVC Disable DatePicker. The problem is telerik mvc datepicker is added in dom using jquery html() function.
After it loaded in dom i have to disable it.
I couln't use "$("#AccountInformalHearingDate").data("tDatePicker").disable()" method.
i'm $("#AccountInformalHearingDate").data("tDatePicker") is undefined error.
  <pre>
          var userId = getUserId();
          $.ajax({
            type: "POST",
            url: $Url.resolve("~/ClientSetup/ClientAccounts/UpdateAccountDetails"),
            data: { AccountId: accountId},
            success: function (data) {
                $('#AccountDetailsContainer').html( @(Html.Telerik().DatePicker()
                    .Name("AccountInformalHearingDate")
                    .Value(new DateTime(2010, 1, 1))

                 if(userId == 10){
                     $("#AccountInformalHearingDate").data("tDatePicker").disable();
                 }

            ));
            },
            error: function (xhr, ajaxOptions, thrownError) {
                alert(xhr.responseText);
            }
        });
  </pre>

But i'm getting control undefined error. Since i cannot disable telerik controls all behaviour without data() function

A: I have found solution for above question after researching telerik mvc documentation
Reason for the above problem is, Ajax returned html contents is not getting reference of registered java scripts. So on success of  ajax call we need manually register needed telerik scripts to access telerik mvc controls as like below"
var telerikScriptArray = new Array();
telerikScriptArray[0] = "/Content/Scripts/Telerik/jquery-1.7.1.min.js";
telerikScriptArray[1] = "/Content/Scripts/Telerik/telerik.common.min.js";
telerikScriptArray[2] = "/Content/Scripts/Telerik/telerik.calendar.min.js";
telerikScriptArray[3] = "/Content/Scripts/Telerik/telerik.datepicker.min.js";
telerikScriptArray[4] = "/Content/Scripts/Telerik/telerik.timepicker.min.js";
telerikScriptArray[5] = "/Content/Scripts/Telerik/telerik.combobox.min.js";

$.ajax({
     type: "POST",
     url: $Url.resolve("~/Ares/Controller/Action"),
     data: { AccountId: accountId},
     sucess:function(data){
         $.telerik.load(telerikScriptArray, function () {
            $("#controlID").data("tDatePicker").disable(); // Here you can access ajax returned telerik controls.
        });
     },
     error:function(){
     }
});

A: Telerik Extensions for ASP.NET MVC Manual
Telerik MVC  Extending the Client API


text:
Home » Members » Governance and Accountability » House of Assembly Management Commission » Hansard Archive » Transcripts
HAMC Minutes
Meeting topics:
Witnesses:
A meeting of the House of Assembly Management Commission was held in the Committee Room, One Government Place, on Tuesday, January 23, 2018 at 5:04 p.m.
Present were: Chair, Hon. Kevin Murphy, Mr Gordon Wilson, Hon. Geoff MacLellan, Mr Chuck Porter, Hon. Labi Kousoulis, Mr Tim Halman, Hon. Chris d'Entremont, Hon. David Wilson and Mr Neil Ferguson, Chief Clerk.
Also present were Gordon D. Hebb, Q.C., Chief Legislative Counsel, Deborah Lusby, Director of Administration, Office of the Speaker and Annette M. Boucher, Q.C., Assistant Clerk.
1) Minutes of the December 12, 2017 meeting
It was moved by Mr Chuck Porter and seconded by Mr Tim Halman that the minutes of the December 12, 2017 meeting be approved as circulated.
2) Appointment of the Auditor General as auditor for 2017-18
Mr Ferguson informed the Commission members that the Audit Committee recommended that the Commission appoint the Auditor General as the auditor for the 2017-18 audit of the House of Assembly.
It was moved by Hon. Labi Kousoulis and seconded by Hon. Chris d'Entremont that the office of the Auditor General be appointed as auditors pursuant to subsection 22(2) of the House of Assembly Management Commission Act for the fiscal year ending March 31, 2018.
3) House of Assembly Management Commission 2017 Annual Report
The 2017 draft report was circulated to Commission members in advance of the meeting.
It was moved by Hon. Geoff MacLellan and seconded by Hon. David Wilson that the 2017 Annual Report of the House of Assembly Management Commission be approved as circulated and be tabled in the House of Assembly by the Speaker.
4) House of Assembly Management Commission Audit Committee 2017 Annual Report
The report was circulated to the members of the Commission in advance of the meeting and was updated since that time to include a summary of the January 17, 2017 Audit Committee meeting that had been inadvertently omitted.
It was moved by Hon. Chris d'Entremont and seconded by Hon. Labi Kousoulis that the 2017 Audit Committee Annual Report be accepted by the Commission.
5) 2017-18 3rd Quarter Financials
Ms Lusby indicated the this 3rd Quarter report reflects forecast adjustments as the budget was approved by the House of Assembly in September 2017. She also pointed out that the PC Caucus and NDP Caucus amounts changed based on the 2017 election results and the change in the number of MLAs.
It was moved by Mr Gordon Wilson and seconded by Mr Tim Halman that the report as presented by accepted.
6) Other
Hon. David Wilson requested that the issue of maternity leave for elected officials be discussed at a future meeting. Ms Lusby will conduct a jurisdictional scan on what the policy is in other provinces. It was suggested that background information also be collected on paternity leave for elected officials. The Chair suggested that this item be placed on the next meeting agenda for an update and discussions.
7) Adjournment
The meeting adjourned at 5:13 p.m.
These minutes were approved by the Management Commission on June 12, 2018.


The last leg of our European adventure brought us to France, specifically the Baux Valley in Provence. Our good friends Nick and Alex were getting married at the gorgeous Chateau d'Estoublon and Avvie was their flower girl. Squee.
We initially thought about driving from Italy, but renting cars between countries can get complicated and very expensive, so we decided to fly instead. Luckily Bologna has a direct flight to Paris and you can catch the TGV (fast train) straight from the airport down to Avignon. From there, it was less than hour to the region. We stayed at Baumaniere, one of the properties Nick and Alex recommended (and which is pictured above). It being a Relais & Chateaux, I knew that I'd probably love it, but their website seriously does not do this place justice. When you think of the South of France, this is probably what you're picturing in your head: beautiful farmhouse buildings scattered amongst manicured gardens, gorgeous pools, and the scent of lavender wafting in the air. And if that's not enough, there are swans, miniature goats, kitties, and a two-Michelin star restaurant.
Our room was in the above building; that's our private terrace on the left. It had its own sitting room as well, which we basically turned into Avvie's room. After four nights of pretty cramped quarters in Bologna, I was ecstatic to not have to whisper after 8PM.
For our first night, since we'd just had a full day of travel, I made us a reservation at the less-formal restaurant on property, La Cabro d'Or. We walked Avvie around the gardens in her stroller right around her bedtime so that she was fast asleep by the time we sat down to eat. We opted for the tasting menu, because ... France, and it was superb: fresh produce from the area, inventive dishes and not too heavy.
The next morning, we took advantage of the incredible breakfast spread at the hotel (fig yogurt! fresh croissants!) and then drove about five minutes to Carrières de Lumières, a series of old stone quarries that now project famous art onto the walls accompanied by music. It's a wonderful spot to cool off and it's also pretty kid-friendly. It definitely gets crowded though, so I'd advise getting there early.
For lunch we took my friend Stephanie's recommendation and drove to Bistrot du Paradou, a highly recommended French restaurant where they only serve one set, four-course menu per day. On Fridays, it's aioli, which, to this American is more of a sauce than a whole entree, but I underestimated how seriously the French love their garlicky mayo.
Observe the aioli. Respect the aioli.
They bring you ALL the cheese and then you eat what you want and they take it away. Dessert was a choice of a bunch of different mousses and puddings, which, if you know me, is by far my favorite category of sweets. We opted for one chocolate and one chestnut, both of which were perfect.
After another perfect meal at Cabro d'Or that night for the rehearsal dinner, the next day was wedding day! It was Hollywood-perfect and Avvie made it like halfway down the aisle before deciding she didn't want to walk anymore, so I'll call that a win.
Obviously if work weren't a thing, we'd be moving here immediately. There's a reason why everyone, both locals and travelers alike, love this region of France. And don't we look just perfect outside this chateau?


text:
/*
 * To change this license header, choose License Headers in Project Properties.
 * To change this template file, choose Tools | Templates
 * and open the template in the editor.
 */
package com.psalgs.list.circular;

/**
 *
 * @author djoshi
 */
public class SortedInsert {

    static class Node {

        int data;
        Node next;

        Node(int d) {
            data = d;
            next = null;
        }
    }

    static Node head;

    private void traverseCList() {
        Node curr = head;

        while (curr != null) {
            System.out.print(curr.data + " ");
            curr = curr.next;

            if (curr == head) {
                break;
            }
        }
        System.out.println();
    }

    private void insert(int d) {
        Node n = new Node(d);
        Node curr = head;
        Node prev = null;

        while (true) {

            if (curr.next != head && curr.data <= d) {
                prev = curr;
                curr = curr.next;
            } else {
                // If node to be inserted is the first node
                if (curr == head) {
                    // traverse to end of list and point last node to new head node
                    while (curr.next != head) {
                        curr = curr.next;
                    }
                    curr.next = n;
                    n.next = head;
                    head = n;
                } else if (curr.next == head) {
                    // node to be inserted is the last node
                    curr.next = n;
                    n.next = head;
                } else {
                    // if node to be inserted is in middle
                    n.next = prev.next;
                    prev.next = n;
                }
                break;
            }
        }

    }

    public static void main(String args[]) {
        SortedInsert llist = new SortedInsert();
        llist.head = new Node(1);
        llist.head.next = new Node(3);
        llist.head.next.next = new Node(5);
        llist.head.next.next.next = new Node(7);
        llist.head.next.next.next.next = new Node(9);
        llist.head.next.next.next.next.next = new Node(11);
        llist.head.next.next.next.next.next.next = head;
        llist.traverseCList();
        System.out.println("Insert in the middle of list");
        llist.insert(8);
        llist.traverseCList();
        System.out.println("Insert at the end of list");
        llist.insert(12);
        llist.traverseCList();
        System.out.println("Insert at the begining of list");
        llist.insert(0);
        llist.traverseCList();
    }
}

redpajama_set_name: RedPajamaGithub


Received 26 national and international awards for research and innovation in cardiac physiology.
Designed innovative medical devices and software.
Andrew L. Rivard, MD, MS is a Staff Physician in the Imaging Institute and Heart and Vascular Institute at Cleveland Clinic Abu Dhabi. In his role as Section Head of Cardiac Imaging, Dr. Rivard is responsible for leading a team of experts in the imaging and diagnosis of cardiovascular disease.
Prior to joining Cleveland Clinic Abu Dhabi, Dr. Rivard spent 5 years at the University of Mississippi where he was an Assistant Professor and Director of Cardiac Imaging. He developed the first program in the State of Mississippi providing complex pediatric congenital heart and adult cardiac imaging using CT and MRI.
Dr. Rivard brings a wealth of knowledge to Abu Dhabi with personalized patient care, organization, and leadership in radiology. His extensive research background includes experience as a principle investigator of multiple large research trials. He has written over 30 peer-reviewed publications in basic science, pre-clinical, and clinical research in wide range of topics including importance of patient-centric care to 3D printing.
Dr. Rivard earned his undergraduate and medical degrees at the University of Minnesota. He also has a master's degree in cardiac physiology from the University of Minnesota. He trained in general surgery at the Geisinger Medical Center and completed a diagnostic radiology residency with a specific focus on cardiac imaging at the University of Florida, Gainesville, USA.
Dr. Rivard resides in Abu Dhabi with his wife and daughter. He enjoys being with his family, sailing in nice weather, cycling, listening to classical music, and traveling to unique destinations.


The arrival slot machine is a space age slot machine with loads of aliens to help you along the way. The main character is a green alien - watch him begin to scold you, if you don't spin those reels!
Arrival is a 3D video slot with a cracking theme tune in the background - very much in the style of the new generation of video slots found in Las Vegas. Arrival slot machine can be played here for free and real money.
If you liked this free Arrival slot machine made by Betsoft, you might like to try some of these popular games which are also available in free-to-play mode.
In addition to that, it is worth knowing that some casinos specialise in certain types of slots play. for example, high roller style players might like to take a look at our high limit slots rooms with games where you can play for real money jackpots.
Also, players in certain countries will have different casinos to choose from. This means that players in the UK should look for UK slots rooms (for Fruit Machines), whilst those in Canada should head to our Canadian casinos section.
Quite a few players are looking for the same games online that they can find in land-based casinos, rather than games like Arrival slots. We have a few, like Cleopatra slots and Monopoly as well as Wheel of Fortune, that you are welcome to play for free.
You can though, visit our Vegas casinos online which lists the casinos that you might enjoy the most. Quite a few new online casinos offer great games, like this Arrival slot machine, which are just as good as the ones you have played in Las Vegas.


Single wooden panel with a simple criss cross pattern. The panel fits around the edge of a decking area for decoration and extra privacy. Overall size: 760 x 1130 x 36mm.
Compatible with the Richard Burbidge Traditional timber panel system.


Whats wrong with the audio on this video as your voice sounds distorted.Cant understand a word.
+Theatre Mad Old video, old encoding. Look for latest videos when looking up how tp's. Video is out of date anyway and won't work for anything but Vista. What you do is the same but how you get there is different. Also there are other things which it could be that I didn't include in this video so.
Will this work for a Windows 8.1 64-bit computer?
By the way if you don't feel safe downloading the torrent there is also another method where you go to Microsoft.com and download the Visual C++.
Well so far it hasn't given me any viruses. And Avast Antivirus hasn't detected any malware so I presume it is safe.
+saad mohamad I am the original up-loader of this video, just on a different channel. I deleted his comment because I don't want people downloading that because there's no way of telling if its safe or not. The only way to check if its clean is to do a registry scan of your computer but....if you didn't already have a scanner installed that had a record of what your registry(at-least the important stuff) looked like before it was altered there's no guarantees. You can also scan the folder you downloaded or do a full computer scan which might see if something was buried inside of it but that won't change the registry back.
You should have stock Microsoft applications like Windows Defender and what not already installed. Just make sure they're turned on and they might catch something if it was changed but really there's nothing you can do...just be careful in the future. Especially with Pirate Bay, if you have no idea what you're downloading or looking for.
+saad mohamad That Torrent you just downloaded could have any kind of worm inside it and you just installed it and allowed it to change things in your registry. NEVER download something like this from 3rd Party, especially because you can get this directly off Microsoft's website. Good luck to you.
+KennyGee Okay never mind then. Plus I managed to solve my problem. Anyways thanks for responding.
will this fix the zmodeler error?
What if my highest number is the same as the "Default" number?
+W. Lynn Then you have a different problem that you probably won't be able to fix without a restore but try this first.
I can't open notepad ._.
+DHWANKSH BHARTI Me neither, If you find the solution to this problem, please share it with me, thanks!
I am running Windows Vista but my default number is the same number as the highest in the list? What can I do?
This idiot sounds like he's got a mouth full of gravel. Can't understand half of what he says. Then, one of the steps he suggests doesn't exist on my Windows7 machine. Utterly useless waste of my time.
+laythisdown Im the idiot yet you are the guy that can't see this isnt for windows 7. I'm the idiot but you are the guy that doesn't know how to work a computer in the age of computers. Keep telling yourself that IM the one thats the idiot in this situation. I'm glad I wasted your time, thanks for the view count and comment you idiot.
Hey i have followed every single step perfectly but i dont know why but when i get to the part where you have to change the value of the default thing i get an error saying that i cant change the value of it, please help me ive been trying to fix this for hours and i cant find a solution.
HEY in my x86_policy.9.0.microsoft.vc90.crt folder there only 3 file in it : Default , and other 2 types and i did the copy paste the hghest number of the type to default then restart and it still not work to open my txdfucker ._.
I don't have components :( how do i fix?
I have win 8.1 I cant get yahoo messenger to install can you help me.
aftter i clicked on x96_policy.9.0.microsoft.vc.90.crt the default did'n't automatically change for me, what do i do?
I am on windows 8 and i do not have winners and I am trying to install smite, what do I do?
This is the video for vista, you probably don't have vista or are looking in the wrong place. worse comes to worse youll have to system restore.
using windows 7 or 8?
Hi - didn't work for me. My default was teh same number as my highest registry number to start with. So then I took it to a pc repair shop. After spending some good money and getting it back fixed, I try to open a pdf and still side by side config. error! I'm running windows vista. what should I try now?
You should have clicked the Windows 7 link in the description and read the comments on the video, I went over all different ways to fix the problem. May have saved you a few dollars. Happy you got your problem resolved though.
Just for sake of helping other out, did they tell you what they did to fix your specific problem?
My Acer 5735 aspire, came up with this "side by side confguration error", and i struggled to overcome it. My brother suggested i try a simple Windows restore, ( start,accessories, system tools, windows restore), and it worked a treat. Up and running again in no time. It's very annoying if you don't have a clue how to fix it and can't access the internet.
+KennyGee Hey it didn't work for me need help.
+thomastidewater gee i was just trying to help out.
Windows 8.1 64 bit for all .exe files and all files except for the ones that came with the computer.
I dont have component for mine,what do i do.
does this fix bully scholarship edition error too??
It doesn't work for me. I'm trying to fix my AVAST 2014 Free Anti-Virus. When I got to the point when I had to edit the registry value, by default it was already set to the largest number.
Didnt work for me. It said "cannot change: had trouble changing new contents values" or something like that. Any ideas on how to fix it?
i don't have that components any helpp???
YOU ARE KING...... I KISS YOU!!!!
Worked for me after some tweaking. First just search for "Winners". It is somewhere else in Windows 7 registry apparently. Go to the event viewer first and see in the side by side error message which policy it is. Not always the same HyoriGene mentioned. Mine was VC80.MFC.
Then find that one and enter the highest number in "Default"
Thanks so much you saved me 8 hours of reinstalling from scratch.
will this work for wheelman?
i have win 7 and i get the side by side error while running eve online will this work?
well, my 9.0 folder only had one binary file in it..
I have only 2 Reg binary ... and the biggest one is allready on the default ... I don`t have what to change ... :| sooo this video it don`t work for me ... and I`m running Windows 7 and I don`t have Components ... I had to go on Software/ Microsoft / Windwos / CurrentVersion/ SidebySide/winners/x86_policy.9.0.microsoft.vc90.crt ... Some one can help me ?
Don't have components I'm very sad.
Woot, it worked directly when i closed regedit on (Vista), you sir are a brilliant lovely man.
Wow!!!!!! Thank you soooo much!
Im doing it for the game called Cities XL Platinum and it keeps popping out that even though i tried every, so please if anyone that has the same problem, helpp!!
I'm sorry guys it won't work for everyone, mostly because alot of you guys are running different operating systems and computer types, the best thing is for you to type fixing side by side configuration error with the name of your OS into google.
Thank you so much for this video, my classes start for school on the 7th and my microsoft office was not working, I followed your steps (which you made very clear and easy to follow) and it fixed my problem...THANK YOU!!!!!!
You've no idea the amount of thanks you sir, deserve.
To everyone else watching this (mostly for gamers) you might have to do this over multiple times after you do it the first time, I know I have to I'm not sure whats up with that but after a few times you wont even need the video for reference. Seems to happen a few weeks apart to a few months.
On computers running Windows 7 or Windows 2008 it will start with "HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows\CurrentVersion\SideBySide\Winners\x86_policy.9.0.microsoft.vc90.crt_".


Acorn Web Design, based in Grove, near Wantage in Oxfordshire, offer web design and search engine optimisation services for small companies and organisations in the Wantage, Faringdon, Hungerford, Didcot, Witney, Abingdon, Oxford area as well as elsewhere in the UK.
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In addition to custom web site design or re-design Acorn Web Design also offer web site hosting, domain name registration and web site maintenance as required - all at competitive rates.
Acorn Web Design is a family business that enjoys working with individuals, start-ups and small businesses and offers a friendly, jargon free service to clients. We can help in all aspects of developing and promoting your web site including assistance with the written content, digital photography and video, scanning, sourcing of stock images, setting up and customising social media pages, etc.
Acorn Web Design are ideally situated for businesses and organisations based in Abingdon, Ardington, Baulking, Blewbury, Charney Bassett, Childrey, Chilton, Denchworth, Didcot, Drayton, East Challow, East Hanney, East Hendred, East Ilsley, East Lockinge, Eastbury, Faringdon, Farnborough, Fawley, Fernham, Frilford, Garford, Goosey, Great Shefford, Grove, Harwell, Hatford, Hungerford, Kingston Bagpuize, Kingston Lisle, Lambourn, Letcombe Bassett, Letcombe Regis, Lyford, Marcham, Milton, Pusey, Shellingford, Sparsholt, Stanford-In-The-Vale, Steventon, Sutton Courtney, Uffington, Wantage, West Challow, West Hanney, West Ilsley and Woolstone.
Although based near Wantage, Oxfordshire and focusing on providing web design services to companies and organisations primarily in Wantage and the surrounding area, Acorn Web Design are also able to offer their web design services to businesses elsewhere in the UK with consultations by email and telephone.


text:
Grădinari è un comune della Romania di 3.233 abitanti, ubicato nel distretto di Giurgiu, nella regione storica della Muntenia.

Il comune è formato dall'unione di 3 villaggi: Grădinari, Tântava, Zorile.

Altri progetti

Collegamenti esterni

Comuni del distretto di Giurgiu


text:
Dave played his guitar alongside his harp and guitar playing brother Louis in The Aces, one of the most influential backing bands in the classic period of Chicago Blues. The brothers came from Mississippi, where they had played local dates with Robert Nighthawk, Sonny Boy Williamson II and Memphis Minnie. They moved to the Windy City in the mid-40s where they began playing house parties with harp player Junior Wells. When Fred Below joined them on drums, they began performing as the Four Aces, which was soon the hottest band in town, except for maybe Muddy Waters outfit. When Muddy's harp player Little Walter wanted to go solo after his chart topping record 'Juke', Junior Wells joined Muddy's band and the Myers boys became The Jukes and backed Little Walter on tour and on record.
Dave Myers Discography
This is Dave's only solo album and his playing is still excellent, as he sings some classics, some Fats Domino and some of his own songs.
YOU CAN'T DO THAT
This arrangement ended in 1955 when they joined up with Otis Rush, who was tearing up the West-side clubs at the time. Dave soon switched to electric bass, and commenced a career as one of the most sought after side-men in the business. The New Aces re-united the Myers brothers with Fred Below, and they enjoyed many years of touring European Festivals and as a premier backing unit for countless Blues stars. In 1998, Dave finally cut an album of his own with 'You Can't Do That' which revisited the glory days of Chess Studios and his early days in the Delta.
Dave plays a lovely instrumental on tour in Argentina;
← LOWELL FULSONEVA TAYLOR →


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text:
Mar 15, 2016 - PlayStation 4, Xbox One, PC, PlayStation Vita, Xbox Series X/S, PlayStation 5
Salt and Sanctuary Media
Salt and Sanctuary Announcement Trailer: Washed Ashore | PS4, PS Vita
Critic Reviews for Salt and Sanctuary
Leif Johnson
You could call Salt and Sanctuary a 2D mashup of Dark Souls and Castlevania, and you'd be right. But such a simple pronouncement would be a disservice to the tremendous amount of thought that's gone into Salt and Sanctuary, whether it's in the graphic style that evokes horror through the style of comic strips or in the intricate web of dungeons and castles serves as it map.
Jeffrey Matulef
Salt and Sanctuary is a derivative Dark Souls clone - but it's a shockingly good one.
If you want a 2D Dark Souls it's hard to imagine From Software doing much better than this, even if it has too few original ideas of its own.
Mashing together the challenge of Dark Souls with the platforming and combat styling of Castlevania makes for a challenging but highly rewarding formula
Miguel Concepcion
This combat-intensive adventure manages to be more than the sum of its Dark Souls-influenced parts.
Salt and Sanctuary lovingly adapts the Souls games into 2D
Nadia Oxford
Salt and Sanctuary is a quality platformer with well-implemented Metroidvania elements. It's brutal at times, but deeply satisfying to play. Is it thematically derivative of the Souls games? Yes, but not disgustingly so. Just get out there and give a few demons what for.
Zack Furniss
It's difficult to be disappointed by a great studio taking solid mechanics and narrative beats from one of the best studios around, especially when the result is something as well-crafted as Salt and Sanctuary. While I do wish Ska Studio's latest had more of its own identity, I can't deny that I enjoyed every minute of it. I already spent about 25 hours with it, and I'm salivating, eager to go through New Game +. Now be proud that I didn't make a "salty" joke like every other outlet will in their review.


What is Kong in Concert?
Kong in Concert is a collaborative arrangement project born from the music of Donkey Kong Country and created for the purposes of reviving, reinterpreting, and rearranging its entire soundtrack with our combined talents; honoring those who originally composed it; and reawakening the nostalgia and love for a landmark videogame.
Why this music and this game?
The Donkey Kong Country soundtracks are different from those of most videogames in that they excel beyond their medium; they are strong in melody, developed in form, and possess sound quality which is impressive to this day. This project concentrates on the first game, whose classic soundtrack was composed by Robin Beanland, Eveline Fischer and substantially Dave Wise; though they each worked on sequels, DKC was their only collaboration. Created by Rare and released by Nintendo on the SNES in 1994, DKC set a new standard with music as rich and colorful as its graphics. Rearranging it without simple regurgitation would be a challenge for any experienced artist, yet each of us found significant room for creative liberty and personal interpretation.
How was this album produced?
The project began April, 2004 when artists were brought together to chose whichever track they wished to arrange, regardless of how many had chosen it. Once all arrangements were complete, contentions were resolved by a peer review in which everyone involved in the project was included. The two competing arrangements deemed by majority vote to be superior were selected for those two tracks (Theme and The Credits Concerto) on the album. After the album was finalized and the project was complete, Kong in Concert was released on September 12th, 2004. This album is in no way commercial and is free to download.


text:
arts-and-entertainment 3
Tac 1
Kong Rithdee 5
Did you mean: department
Saint and sensibility
Life, Kong Rithdee, Published on 16/05/2019
» A Christian fable or a Marxist allegory? A magical-realist myth or a political cry against neoliberalism (or feudalism, which produces the same catastrophe anyway)?
EU film fest brings many shades of modern Europe
» The stories of Europe are told in the 13 films at the European Union Film Festival 2017, which begins tonight at SF CentralWorld.
» With over 400 movies on the slot, the Toronto International Film Festival was a feast and a maze. The latest edition of this North American showcase concluded last Sunday, with Damein Chazelle's La La Land winning the People's Choice Award, a bellwether for the bright Oscar season (Toronto, unlike other major festivals, has no prominent juried competition, instead letting the audiences decide the big winner). The festival is known as a launch pad for Oscar hopefuls as well as independent titles looking for distribution. It also features a strong experimental section that casts its radical net far and wide.
Don't they know it's 'our' Songkran?
News, Kong Rithdee, Published on 22/03/2014
» Your Honour, ladies and gentlemen of the tribunal of the International Court of Universal Justice. I've come to this court today to represent the Cultural Watchdog and Tourism Authority of Thailand, and I henceforth present to the esteemed tribunal the grave, scandalous, hair-raising offence committed by a group of people in Singapore (I think) against the splendid and extremely wet Siamese tradition known as Songkran.
Silence, please!
» Silence has now joined the Bengal tiger on the list of endangered species.


the term Gabelle or Gabella (of the medieval of latin gabulum) was used in France originally for taxes on any kind by goods, was limited however gradually to the salt duty. As gabella emigrationis it was a departure money, which was to be paid in former times the fortune carried forward by an emigrant for. Gabella hereditatis was the tax, which was to be paid for an inheritance or a donation going abroad.
the Pays exempts, which required an exception of the tax, when they were united with the Kingdom of France.
The producers had to supply their salt to the Greniers à sel (salt chambers since 1342), which were furnished in each province; during neglect Konfiszierung threatened. The salt chamber determined the price, it for the salt paid and sold it then to a higher course to the retailers.


The Central Government on Monday extended the ceasefire which has been in operation, with GPRN/NSCN and the NSCN (Reformation) for another year.
A report quoting Home Ministry sources further stated that a fresh pact for suspension of operations was signed with NSCN/GPRN (K) headed by Khango Konyak.
Both the agreements were signed on Monday by MHA joint secretary Satyendra Garg on behalf of the Union Government.
GPRN/NSCN supervisor Jack Jimomi signed on behalf of GPRN/NSCN and the signatories for NSCN (R) were supervisor Dr Amento Chishi and secretary Toshi Longkumer, reports added.
Later in communiqué issued to the media, GPRN/NSCN through its MIP stated that after thorough deliberation, both the sides decided to uphold the agreed ceasefire ground rules in letter and in spirit.
Meanwhile, NSCN/GPRN (K) also entered into a 'fresh ceasefire agreement with the Centre from Monday for a period of one year with effect from April 15, 2019,' the Ministry of Home Affairs had stated.
The new ceasefire agreement and revised ceasefire ground rules were signed by Centre's representative- MHA joint secretary Satyendra Garg and NSCN/GPRN (K) representatives-supervisor Newell Naga and member Michael Yeptho.
It may be recalled that after the split of NSCN (K) in 2018, NSCN/GPRN (K) led by Khango Konyak expressed desire to restore the ceasefire.
On December 17 last year, the group held an 'informal meeting' with the Central Government in New Delhi to chalk out the differences that had cropped up after abrogation of the cease-fire in March 2015.
The meeting was held following NSCN/GPRN (K) led by Khango on December 6 last year declared to revoke its decision of unilaterally abrogating the ceasefire agreement with the Union Government.


With the help of excellent teachers and gifted book-making professionals, we publish select student work created in the Bureau of Fearless Ideas workshops and during our publishing projects. Our anthologies are on sale in our store and online. All proceeds from the sale of books (and space travel supplies) go to fund our programs.
Our third anthology of work produced by Hamilton International Middle School students and 826 Seattle workshop participants.
Our second anthology of work produced by Hamilton International Middle School students explores the theme of family.
This inventive anthology of comics by middle and high school students is the result of a series of comics writing workshops taught by local artists David Lasky and Jessixa Bagley.
Our final collaboration with John Marshall Alternative High School, which closed its doors in June of this year, is a poignant collection of stories on the theme of "happiness," which are often far from happy.
Our first of two anthologies this year with the students at Hamilton International Middle School. For the last five months, 826 Seattle volunteers and staff have been working with the teachers and students at Hamilton to get these stories on paper and the results are hilarious, heartbreaking and inspiring.
Complied from summer workshops and our writing contest from last year, these stories are a little slice of the Pacific Northwest as seen through the eyes of 826 Seattle students.
Seattle comic book artist David Lasky taught a comics class in the spring of 2007 to a group of 12 to 17 year olds and the results are simply spectacular. Eleven students are published in this collection of comics about family.