text:
\section*{Introduction}

Throughout this paper, by a \emph{Hankel matrix}, we mean a matrix of the form
\[
H\colonequals\begin{pmatrix}
x_1 & x_2 & x_3 & \cdots & x_s\\
x_2 & x_3 & \cdots & \cdots & x_{s+1}\\
x_3 & \cdots & \cdots & \cdots & x_{s+2}\\
\vdots & \vdots & \vdots & \vdots & \vdots\\
x_r & \cdots & \cdots & \cdots & x_{s+r-1}
\end{pmatrix},
\]
where $x_1,\dots,x_{s+r-1}$ are indeterminates over a field $\mathbb{F}$. By a \emph{Hankel determinantal ring} we mean a ring of the form
\[
\mathbb{F}[x_1,\dots,x_{s+r-1}]/I_t(H),
\]
where~$1\leqslant t\leqslant\min\{r,s\}$, and $I_t(H)$ is the ideal generated by the size $t$ minors of~$H$. These rings arise as homogeneous coordinate rings of higher order secant varieties of rational normal curves, see for example Room's 1938 study \cite[Chapter~11.7]{Room}.

We prove that Hankel determinantal rings over fields of characteristic zero have rational singularities, Theorem~\ref{theorem:ratsing}. In particular, higher order secant varieties of rational normal curves have rational singularities. Theorem~\ref{theorem:ratsing} may be compared with corresponding statements for generic determinantal rings, and those defined by minors of symmetric matrices of indeterminates or by pfaffians of skew-symmetric matrices of indeterminates: in characteristic zero, these are all invariant rings of linearly reductive classical groups acting on polynomial rings, and hence are pure subrings of polynomial rings. By Boutot's theorem~\cite{Boutot}, it then follows that they have rational singularities. We do not know if Hankel determinantal rings, in general, arise as invariant rings for group actions on polynomial rings, or if they are pure subrings of polynomial rings. However, for $t\geqslant 3$, we show that they are not pure subrings of the polynomial rings in which they are naturally embedded, see Proposition~\ref{proposition:not:pure:subrings}. Our proof of rational singularities is via reduction modulo~$p$ methods, using Smith's theorem \cite{Smith:ratsing} that rings of $F$-rational type have rational singularities.

We compute the divisor class groups of Hankel determinantal rings: the group is finite cyclic, in particular, the rings are $\mathbb{Q}$-Gorenstein, and we show that each divisor class group element corresponds to a rank one maximal Cohen-Macaulay module, see Theorem~\ref{theorem:class:group}.

We also prove that Hankel determinantal rings over fields of positive characteristic are~$F$-pure, Theorem~\ref{theorem:fpure}. Finally, for $R$ a Hankel determinantal ring with homogeneous maximal ideal $\mathfrak{m}_R$, we compute the $F$-pure threshold of~$\mathfrak{m}_R$ in $R$, and of its defining ideal~$I_t(H)$ in the ambient polynomial ring, see Theorems~\ref{theorem:fpt1} and~\ref{theorem:fpt2}.

\section{Generalities}

By a result of Gruson and Peskine,~\cite[Lemme~2.3]{Gruson-Peskine}, every Hankel determinantal ring is isomorphic to one where the defining ideal $I_t(H)$ is generated by the maximal sized minors of a Hankel matrix; alternatively see \cite[Proposition~7]{JWatanabe} or \cite[Corollary~2.2(b)]{Conca}. In view of this, we will henceforth work with Hankel determinantal rings of the form
\[
R\colonequals\mathbb{F}[x_1,\dots,x_{n+t-1}]/I_t(H),
\]
where $H$ is a $t\times n$ Hankel matrix and $t\leqslant n$; except where stated otherwise, $H$ will denote such a~matrix.

Consider the \emph{generic} determinantal ring
\[
B\colonequals\mathbb{F}[Y]/I_t(Y),
\]
where $Y$ is a~$t\times n$ matrix of indeterminates, and $I_t(Y)$ the ideal generated by its size $t$ minors. The $(t-1)(n-1)$ elements $Y_{i,j+1}-Y_{i+1,j}$ are readily seen to be part of a system of parameters for $B$, and specializing these to $0$ gives a ring isomorphic to~$R$. Since~$B$ is Cohen-Macaulay by \cite{Eagon-Northcott, Hochster-Eagon}, so is the ring $R$, and the Eagon-Northcott complex provides a minimal free resolution of $R$. It follows as well that
\[
\dim R\ =\ 2t-2,
\]
and hence that
\[
\operatorname{height}\, I_t(H)\ =\ n-t+1.
\]
The elements $x_1,\dots,x_{t-1},x_{n+1},\dots,x_{n+t-1}$ are a homogeneous system of parameters for~$R$, and the socle modulo this system of parameters is spanned by the degree $t-1$ monomials in~$x_t,\dots,x_n$. In particular, the ring $R$ has $a$-invariant
\[
a(R)\ =\ 1-t.
\]
The multiplicity of the ring $R$ is
\[
e(R)\ =\ \binom{n}{t-1},
\]
as may be seen directly from the above discussion, or obtained using the multiplicity of generic determinantal rings, e.g., \cite[Proposition~2.15]{Bruns-Vetter}.

The ring $R$ is a normal domain, see for example,~\cite[Proposition~8]{JWatanabe}; it is Gorenstein precisely when~$t=n$. The ideal $I_t(H)$ is a set-theoretic complete intersection by~\cite{Valla}. The singular locus of $R$ is defined by the image of $I_{t-1}(H)$, see \cite[Theorem~1.56]{Iarrobino-Kanev}. For secant varieties of smooth curves in general, we mention \cite{Vermeire} and the references therein.

\begin{notation}
Given a matrix $X$, we use ${[a_1\ \dots\ a_r\mid b_1\ \dots\ b_r]}_X$ to denote the determinant of the submatrix of $X$ with rows $a_1,\dots,a_r$ and columns $b_1,\dots,b_r$. We omit the subscript whenever the matrix is clear from the context.
\end{notation}

\section{Rational singularities}

In proving that Hankel determinantal rings of characteristic zero have rational singularities, we will use the following description: A $2\times n$ Hankel determinantal ring over a field~$\mathbb{F}$ is readily seen to be isomorphic to the $n$-th Veronese subring of a polynomial ring~$\mathbb{F}[u,v]$, where the Hankel matrix maps entrywise to
\[
\begin{pmatrix}
u^n & u^{n-1}v & u^{n-2}v^2 & \cdots & uv^{n-1}\\
u^{n-1}v & u^{n-2}v^2 & \cdots & \cdots & v^n
\end{pmatrix}.
\]
This is the homogeneous coordinate ring of the rational normal curve $C_n$ in $\mathbb{P}^n$; as it is a Veronese subring, it is a pure subring of $\mathbb{F}[u,v]$, independent of the characteristic of $\mathbb{F}$.

A $3\times n$ Hankel determinantal ring is the homogeneous coordinate ring of the secant variety of the rational normal curve $C_{n+1}$ in $\mathbb{P}^{n+1}$; it is isomorphic to the subring of the polynomial ring $\mathbb{F}[u_1,u_2,v_1,v_2]$, where the Hankel matrix maps entrywise to the matrix
\[
\begin{pmatrix}
u_1^{n+1}+u_2^{n+1} & u_1^nv_1^{\phantom.}+u_2^nv_2^{\phantom.} & u_1^{n-1}v_1^2+u_2^{n-1}v_2^2 & \cdots & u_1^2v_1^{n-1}+u_2^2v_2^{n-1}
\\[0.4em]
u_1^nv_1^{\phantom.}+u_2^nv_2^{\phantom.} & u_1^{n-1}v_1^2+u_2^{n-1}v_2^2 & \cdots & \cdots & u_1^{\phantom.}v_1^n+u_2^{\phantom.}v_2^n
\\[0.4em]
u_1^{n-1}v_1^2+u_2^{n-1}v_2^2 & \cdots & \cdots & \cdots & v_1^{n+1}+v_2^{n+1}
\end{pmatrix}.
\]

More generally, the Hankel determinantal ring $R\colonequals\mathbb{F}[x_1,\dots,x_{n+t-1}]/I_t(H)$ is the homogeneous coordinate ring of the order $t-2$ secant variety of the rational normal curve~$C_{n+t-2}$ in $\mathbb{P}^{n+t-2}$, see for example~\cite[Section~4]{Eisenbud}. Specifically, we claim that $R$ is isomorphic to the $\mathbb{F}$-subalgebra of the polynomial ring
\[
S\colonequals\mathbb{F}[u_1,\dots,u_{t-1},v_1,\dots,v_{t-1}]
\]
generated by the elements
\begin{equation}
\label{equation:secant}
h_i\ \colonequals\ u_1^{n+t-2-i}v_1^i\ +\ u_2^{n+t-2-i}v_2^i\ +\ \cdots\ +\ u_{t-1}^{n+t-2-i}v_{t-1}^i,
\quad\text{ for }\ 0\leqslant i\leqslant n+t-2.
\end{equation}

To see this, consider the $\mathbb{F}$-algebra homomorphism $\varphi\colon\mathbb{F}[x_1,\dots,x_{n+t-1}]\longrightarrow S$ defined by $\varphi(x_i)=h_{i-1}$ for each $i$. Note that $\varphi$ maps the Hankel matrix of indeterminates $H$ to a matrix $M$ that is Hankel in the elements~$h_i$. As $M$ is the sum of $t-1$ matrices of the form
\[
\begin{pmatrix}
u^{n+t-2} & u^{n+t-3}v & u^{n+t-4}v^2 & \cdots & u^{t-1}v^{n-1}\\
u^{n+t-3}v & u^{n+t-4}v^2 & \cdots & \cdots & u^{t-2}v^n\\
u^{n+t-4}v^2 & \cdots & \cdots & \cdots & u^{t-3}v^{n+1}\\
\vdots & \vdots & \vdots & \vdots & \vdots\\
u^{n-1}v^{t-1} & \cdots & \cdots & \cdots & v^{n-t+2}
\end{pmatrix},
\]
each having rank~$1$, it follows that the rank of $M$ is at most $t-1$, i.e., that $I_t(M)=0$. Hence~$\varphi$ induces a homomorphism $\widetilde{\varphi}\colon R\longrightarrow S$. Since $R$ is a domain of dimension $2t-2$, which is also the dimension of $S$, it follows that $\widetilde{\varphi}$ is injective.

\begin{theorem}
\label{theorem:ratsing}
Let $R=\mathbb{F}[x_1,\dots,x_{n+t-1}]/I_t(H)$, where $\mathbb{F}$ is a field, and $H$ is a $t\times n$ Hankel matrix. If $\mathbb{F}$ has characteristic zero, then $R$ has rational singularities. If $\mathbb{F}$ is a field of positive characteristic $p$, with $p\geqslant t$, then $R$ is $F$-rational.
\end{theorem}

It follows that if $\mathbb{F}$ has characteristic $p\geqslant t$, then the ring $R$ has rational singularities in the sense of \cite[Definition~1.3]{Kovacs}; see \cite[Corollary~1.12]{Kovacs}.

\begin{proof}
It suffices to prove the positive characteristic assertion in the theorem: it then follows that $R$ is of $F$-rational type for $\mathbb{F}$ of characteristic zero, and then by \cite[Theorem~4.3]{Smith:ratsing} that $R$ has rational singularities.

Let $\mathbb{F}$ be a field of characteristic $p\geqslant t$, and assume $t\geqslant 2$. Using \cite[Theorem~4.7]{HH:JAG} and the preceding remark in that paper, it suffices to prove that the ideal generated by one choice of a homogeneous system of parameters for~$R$ is tightly closed. Set
\[
S\colonequals\mathbb{F}[u_1,\dots,u_{t-1},v_1,\dots,v_{t-1}],
\]
i.e., $S$ is a polynomial ring in $2t-2$ indeterminates, and identify $R$ with the subring generated by the elements $h_0,\dots,h_{n+t-2}$ as in~\eqref{equation:secant}. The elements
\[
h_0,\dots,h_{t-2},h_n,\dots,h_{n+t-2}
\]
form a homogeneous system of parameters for~$R$. Let $\mathfrak{a}$ be the ideal of $R$ generated by these elements; it suffices to show that~$\mathfrak{a}$ is tightly closed. Note that $h_i$ belongs to the ideal~$(u_1^n,u_2^n,\dots,u_{t-1}^n)S$ for $0\leqslant i\leqslant t-2$, and to~$(v_1^n,v_2^n,\dots,v_{t-1}^n)S$ for $n\leqslant i\leqslant n+t-2$, so
\[
\mathfrak{a} S\ \subseteq\ (u_1^n,u_2^n,\dots,u_{t-1}^n,v_1^n,v_2^n,\dots,v_{t-1}^n)S.
\]

The socle of $R/\mathfrak{a}$ is the vector space spanned by the images of the elements
\[
h_{i_1}h_{i_2}\cdots h_{i_{t-1}}
\quad\text{ where }\ t-1\leqslant i_1\leqslant i_2\leqslant\cdots\leqslant i_{t-1}\leqslant n-1.
\]
Suppose that a linear combination of the above elements, say
\[
r\colonequals\sum\lambda_{i_1 i_2 \cdots i_{t-1}} h_{i_1}h_{i_2}\cdots h_{i_{t-1}}
\quad\text{ where }\ \lambda_{i_1 i_2 \cdots i_{t-1}}\in\mathbb{F},
\]
belongs to $\mathfrak{a}^*$, i.e., to the tight closure of $\mathfrak{a}$ in $R$. Since $R\subset S$ is an inclusion of domains, it then follows from the definition of tight closure that $r\in (\mathfrak{a} S)^*$. But $(\mathfrak{a} S)^*=\mathfrak{a} S$ since $S$ is regular, implying that $r\in\mathfrak{a} S$, and hence that
\[
r\ \in\ (u_1^n,u_2^n,\dots,u_{t-1}^n,v_1^n,v_2^n,\dots,v_{t-1}^n)S.
\]
We claim that this occurs only when each coefficient $\lambda_{i_1 i_2 \cdots i_{t-1}}$ equals $0$; it then follows that~$r=0$, i.e., that $\mathfrak{a}$ is tightly closed, as desired.

We first illustrate the proof of the claim when $t=3$. In this case, the ring $R$ may be identified with the $\mathbb{F}$-subalgebra of $S=\mathbb{F}[u_1,u_2,v_1,v_2]$ generated by the elements
\[
h_i\ =\ u_1^{n+1-i}v_1^i + u_2^{n+1-i}v_2^i
\quad\text{ where }\ 0\leqslant i\leqslant n+1.
\]
Suppose
\[
r\ =\sum_{2\leqslant i_1\leqslant i_2\leqslant n-1} \lambda_{i_1i_2} h_{i_1}h_{i_2}\ \in\ (u_1^n,u_2^n,v_1^n,v_2^n)S.
\]
Fix $k_1,k_2$ with $2\leqslant k_1\leqslant k_2\leqslant n-1$, and consider the coefficient of $u_1^{n+1-k_1}v_1^{k_1} u_2^{n+1-k_2}v_2^{k_2}$ in the expression above, i.e., in
\[
\sum\lambda_{i_1i_2} h_{i_1}h_{i_2}\ =\ \sum\lambda_{i_1i_2} (u_1^{n+1-i_1}v_1^{i_1} + u_2^{n+1-i_1}v_2^{i_1}) (u_1^{n+1-i_2}v_1^{i_2} + u_2^{n+1-i_2}v_2^{i_2}).
\]
This coefficient is $\lambda_{k_1k_2}$ if $k_1<k_2$, and it equals $2\lambda_{k_1k_1}$ if $k_1=k_2$. Since the characteristic of~$\mathbb{F}$ is~$p\geqslant 3$, and $r\in (u_1^n,u_2^n,v_1^n,v_2^n)S$, it follows that each coefficient must be $0$ as claimed.

We now turn to the general case: suppose
\[
r\ =\sum\lambda_{i_1 i_2 \cdots i_{t-1}} h_{i_1}h_{i_2}\cdots h_{i_{t-1}}\ \in\ (u_1^n,u_2^n,\dots,u_{t-1}^n,v_1^n,v_2^n,\dots,v_{t-1}^n)S,
\]
where the sum is over indices with $t-1\leqslant i_1\leqslant i_2\leqslant\cdots\leqslant i_{t-1}\leqslant n-1$. Let $k_1,\dots,k_{t-1}$ be integers with
\[
t-1\leqslant k_1\leqslant k_2\leqslant\cdots\leqslant k_{t-1}\leqslant n-1.
\]
The coefficient of
\[
u_1^{n+t-2-k_1}v_1^{k_1} u_2^{n+t-2-k_2}v_2^{k_2} \cdots u_{t-1}^{n+t-2-k_{t-1}}v_{t-1}^{k_{t-1}}
\]
in $r$ is $c\lambda_{k_1 k_2 \cdots k_{t-1}}$ where $c$ is a product of positive integers, each less than $t$. Hence $c\neq0$ in~$\mathbb{F}$, and so it follows that each coefficient is $0$.
\end{proof}

While the description in terms of higher secant varieties shows that every Hankel determinantal ring is a subring of a polynomial ring, it is not in general a pure subring of that polynomial ring, as we show next; recall that a ring homomorphism $R\longrightarrow S$ is \emph{pure} if
\[
R\otimes_RM\longrightarrow S\otimes_RM
\]
is injective for each $R$-module $M$.

\begin{proposition}
\label{proposition:not:pure:subrings}
Let $R$ be a $t\times n$ Hankel determinantal ring, regarded as the $\mathbb{F}$-subalgebra of the polynomial ring $S=\mathbb{F}[u_1,\dots,u_{t-1},v_1,\dots,v_{t-1}]$, generated by the elements $h_i$ as in~\eqref{equation:secant}. If $t\geqslant 3$, then $R$ is not a pure subring of~$S$.
\end{proposition}

\begin{proof}
Let $\mathfrak{m}_R$ denote the homogeneous maximal ideal of $R$. The expansion of this ideal to $S$ is contained in the height $t$ ideal
\[
(u_1-v_1,\ \dots,\ u_{t-1}-v_{t-1},\ v_1^{n+t-2}+\cdots+v_{t-1}^{n+t-2})S.
\]
Since $\operatorname{height}\,\mathfrak{m}_RS\leqslant t<2t-2=\dim S$, the Hartshorne-Lichtenbaum Vanishing Theorem, for example \cite[Theorem~14.1]{24hours}, implies that
\[
H^{2t-2}_{\mathfrak{m}_R}(S)\ =\ H^{2t-2}_{\mathfrak{m}_RS}(S)\ =\ 0.
\]
If $R\longrightarrow S$ is pure, the injectivity of $H^{2t-2}_{\mathfrak{m}_R}(R)\longrightarrow H^{2t-2}_{\mathfrak{m}_R}(S)$ implies that $H^{2t-2}_{\mathfrak{m}_R}(R)=0$, which is a contradiction since $\dim R=2t-2$.
\end{proof}

\begin{remark}
\label{remark:not:pure:subrings}
Being a pure subring of a polynomial ring is a stronger property than having rational singularities, or even having $F$-regular type; the hypersurface in~\cite[Theorem~5.1]{Singh-Swanson} has $F$-regular type, but is not a pure subring of a polynomial ring.
\end{remark}

\begin{question}
\label{question:pure:subrings}
Is every Hankel determinantal ring a pure subring of a polynomial ring?
\end{question}

\section{The divisor class group}

Consider the Hankel determinantal ring $R=\mathbb{F}[x_1,\dots,x_{n+t-1}]/I_t(H)$, where $\mathbb{F}$ is a field. To avoid some trivialities, we assume throughout this section that $n\geqslant t\geqslant 2$. Set $\mathfrak{p}$ to be the ideal of $R$ generated by the maximal minors of the first $t-1$ rows of $H$, i.e.,
\begin{equation}
\label{equation:p}
\mathfrak{p}\colonequals I_{t-1}\begin{pmatrix}
x_1 & x_2 & x_3 & \cdots & x_n\\
x_2 & x_3 & \cdots & \cdots & x_{n+1}\\
x_3 & \cdots & \cdots & \cdots & x_{n+2}\\
\vdots & \vdots & \vdots & \vdots & \vdots\\
x_{t-1} & \cdots & \cdots & \cdots & x_{n+t-2}
\end{pmatrix}.
\end{equation}
The ring $R/\mathfrak{p}$ may be identified with the polynomial ring in the indeterminate $x_{n+t-1}$ over a size $(t-1)\times n$ Hankel determinantal ring; it follows that $R/\mathfrak{p}$ is an integral domain of dimension $2t-3$, and hence that $\mathfrak{p}$ is a prime ideal of height $1$.

For each integer $k$ with $1\leqslant k\leqslant n-t+2$, set $\mathfrak{p}^{\langle k\rangle}$ to be the ideal of $R$ as below,
\[
\mathfrak{p}^{\langle k\rangle}\colonequals I_{t-1}\begin{pmatrix}
x_1 & x_2 & x_3 & \cdots & x_{n-k+1}\\
x_2 & x_3 & \cdots & \cdots & x_{n-k+2}\\
x_3 & \cdots & \cdots & \cdots & x_{n-k+3}\\
\vdots & \vdots & \vdots & \vdots & \vdots\\
x_{t-1} & \cdots & \cdots & \cdots & x_{n-k+t-1}
\end{pmatrix}.
\]
Note that $\mathfrak{p}^{\langle 1\rangle}=\mathfrak{p}$, and that the ideal $\mathfrak{p}^{\langle n-t+2\rangle}$ is principal. With this notation, we prove:

\begin{theorem}
\label{theorem:class:group}
Consider the Hankel determinantal ring $R\colonequals\mathbb{F}[x_1,\dots,x_{n+t-1}]/I_t(H)$, for~$\mathbb{F}$ a field, and $t\geqslant 2$. Then the divisor class group of $R$ is cyclic of order $n-t+2$, generated by the ideal $\mathfrak{p}$ as in~\eqref{equation:p}. The symbolic powers of $\mathfrak{p}$ are
\[
\mathfrak{p}^{(k)}=\mathfrak{p}^{\langle k\rangle}
\quad\text{ for }\ 1\leqslant k\leqslant n-t+2.
\]
Moreover, each of these is a maximal Cohen-Macaulay $R$-module.
\end{theorem}

We need a number of preliminary results.

\begin{lemma}
\label{lemma:identity}
Let $Y$ be an $m\times n$ matrix with entries in a commutative ring. Assume that $Y$ has rank less than $t$.

\begin{enumerate}[\ \rm(1)]
\item For every choice of row and column indices, one has
\begin{multline*}
[a_1\ \dots\ a_{t-1} \mid b_1\ \dots\ b_{t-1}] \times [c_1\ \dots\ c_{t-1} \mid d_1 \ \dots\ d_{t-1}] \\
=\
[a_1\ \dots\ a_{t-1} \mid d_1\ \dots\ d_{t-1}] \times [c_1\ \dots\ c_{t-1} \mid b_1 \ \dots\ b_{t-1}].
\end{multline*}

\item Let $Y(a,b)$ denote the submatrix of $Y$ with row indices $\leqslant a$, and column indices $\leqslant b$. Then, for all $a<m$ and $b<n$, one has
\[
I_{t-1}(Y(a,b+1))\ I_{t-1}(Y(a+1,b))\ =\ I_{t-1}(Y(a,b))\ I_{t-1}(Y(a+1,b+1)).
\]
\end{enumerate}
\end{lemma}

\begin{proof}
First note that (2) follows immediately from (1). To prove (1), we may assume right away that the underlying ring is $B\colonequals\mathbb{Z}[X]/I_t(X)$, with $X$ an $m\times n$ matrix of indeterminates, and that $Y$ is the image of $X$ in $B$. Since $B$ is a domain, it suffices to verify the displayed identity in the fraction field $\mathbb{K}$ of $B$. Consider the linear map
\[
\varphi\colon\mathbb{K}^{m}\longrightarrow\mathbb{K}^{n}
\]
given by the image of $Y$. The map $\varphi$ has rank less than $t$, so the exterior power
\[
\Lambda^{t-1}\varphi\colon\Lambda^{t-1} \mathbb{K}^{m}\longrightarrow\Lambda^{t-1} \mathbb{K}^{n}
\]
is a linear map of rank at most $1$. For rows $i_1,\dots,i_{t-1}$ and columns $j_1,\dots,j_{t-1}$ of $\varphi$, the corresponding matrix entry of $\Lambda^{t-1} \varphi$ is the determinant
\[
{[i_1\ \dots\ i_{t-1} \mid j_1\ \dots\ j_{t-1}]}_Y.
\]
The required identity is now immediate from the fact that the size $2$ minors of the matrix for $\Lambda^{t-1}\varphi$ are zero.
\end{proof}

\begin{lemma}
\label{lemma:valuation}
Let $\mathfrak{p}$ be as in~\eqref{equation:p}, and let $v$ denote the valuation of the discrete valuation ring~$R_\mathfrak{p}$. Then, for integers $1\leqslant i_1<i_2<\cdots<i_{t-1}\leqslant n$, the minors of $H$ satisfy
\[
v([1\ \dots\ t-1 \mid i_1\ \dots\ i_{t-1}])\ =\ n+1-i_{t-1}.
\]
Consequently for $\mathfrak{p}^{\langle k\rangle}$ as defined earlier, and $k$ with $1\leqslant k\leqslant n-t+2$, one has
\[
\mathfrak{p}^{\langle k\rangle}\subseteq\mathfrak{p}^{(k)}
\quad\text{and}\quad
\mathfrak{p}^{\langle k\rangle}R_\mathfrak{p}=\mathfrak{p}^{(k)}R_\mathfrak{p}.
\]
\end{lemma}

\begin{proof}
Set $\pi\colonequals [1\ \dots\ t-1\mid n-t+2\ \dots\ n]$. We will prove inductively that
\begin{equation}
\label{equation:val1}
v([1\ \dots\ t-1 \mid i_1\ \dots\ i_{t-1}])\ =\ (n+1-i_{t-1})v(\pi),
\end{equation}
with the base case for the induction being $i_{t-1}=n$. By Lemma~\ref{lemma:identity}~(1) one has
\begin{equation}
\label{equation:val2}
[1\ \dots\ t-1 \mid i_1\ \dots\ i_{t-1}] \times [2\ \dots\ t \mid n-t+2\ \dots\ n]
\ =\ \pi\times [2\ \dots\ t \mid i_1\ \dots\ i_{t-1}].
\end{equation}
We work in the ring $R_\mathfrak{p}$, where the minor
\[
[2\ \dots\ t \mid n-t+2\ \dots\ n]
\]
is a unit. If~$i_{t-1}=n$, then $[2\ \dots\ t \mid i_1\ \dots\ i_{t-1}]$ is a unit in $R_\mathfrak{p}$ as well, and it follows that
\[
v([1\ \dots\ t-1 \mid i_1\ \dots\ i_{t-1}])\ =\ v(\pi),
\]
which proves the base case. For the inductive step, assume that $i_{t-1}<n$ and that~\eqref{equation:val1} holds for larger values of $i_{t-1}$. Since
\[
[2\ \dots\ t \mid i_1\ \dots\ i_{t-1}]\ =\ [1\ \dots\ t-1 \mid i_1+1\ \dots\ i_{t-1}+1],
\]
the inductive hypothesis gives
\[
v([2\ \dots\ t \mid i_1\ \dots\ i_{t-1}])\ =\ (n-i_{t-1})v(\pi).
\]
Combining this with~\eqref{equation:val2}, it follows that
\[
v([1\ \dots\ t-1 \mid i_1\ \dots\ i_{t-1}])\ =\ v(\pi)+(n-i_{t-1})v(\pi)\ =\ (n+1-i_{t-1})v(\pi),
\]
which completes the proof of~\eqref{equation:val1}.

Since the valuation of each minor generating the ideal $\mathfrak{p}$ is a positive integer multiple of~$v(\pi)$, it follows that $\pi$ generates the maximal ideal of $R_\mathfrak{p}$, and that $v(\pi)=1$. Lastly, note that the minors that generate the ideal~$\mathfrak{p}^{\langle k\rangle}$ are precisely those with valuation at least $k$.
\end{proof}

The following is a slight modification of \cite[Lemma~4]{JWatanabe}, adapted to our notation, and with a shorter proof.

\begin{lemma}
\label{lemma:watanabe}
Let $R$ be a $t\times n$ Hankel determinantal ring over a field $\mathbb{F}$. Set
\[
\Delta\colonequals[1\ \dots\ t-1 \mid 1\ \dots\ t-1],
\]
viewed as an element of $R$. Then:
\begin{enumerate}[\ \rm(1)]
\item the ideal $\Delta R$ has radical $\mathfrak{p}$, for $\mathfrak{p}$ as in~\eqref{equation:p},
\item $R_\Delta = {\mathbb{F}[x_1,\ \dots,\ x_{2t-2}}]_\Delta$, and
\item the elements $x_1,\ \dots,\ x_{2t-2}$ of $R$ are algebraically independent over~$\mathbb{F}$.
\end{enumerate}
\end{lemma}

\begin{proof} (1) In the notation of Lemma~\ref{lemma:identity}, the ideal $\mathfrak{p}^{\langle k\rangle}$ is $I_{t-1}(Y(t-1,n-k+1))$, where~$Y$ is the image of the Hankel matrix $H$ in $R$. Since
\[
I_{t-1}(Y(t-1,n-k+1))\ =\ I_{t-1}(Y(t,n-k)),
\]
Lemma~\ref{lemma:identity}~(2) gives
\begin{multline*}
I_{t-1}(Y(t-1,n-k+1))^2\ =\ I_{t-1}(Y(t-1,n-k))\ I_{t-1}(Y(t,n-k+1))\\
\subset \ I_{t-1}(Y(t-1,n-k))
\end{multline*}
i.e.,
\[
(\mathfrak{p}^{\langle k\rangle})^2\ \subset\ \mathfrak{p}^{\langle k+1\rangle}.
\]
Since $\mathfrak{p}^{\langle 1\rangle}=\mathfrak{p}$ and $\mathfrak{p}^{\langle n-t+2\rangle}=\Delta R$, we are done.

(2) For each $a\geqslant t$, we have $[1\ \dots\ t \mid 1\ \dots\ t-1 \ a]=0$ in $R$, so
\[
x_{t+a-1}\Delta\ \in\ \mathbb{F}[x_1, \dots, x_{t+a-2}].
\]
Since $\Delta\in\mathbb{F}[x_1, \dots, x_{t+a-2}]$, it follows that
\[
{\mathbb{F}[x_1, \dots, x_{t+a-1}]}_\Delta\ =\ {\mathbb{F}[x_1, \dots, x_{t+a-2}]}_\Delta.
\]
Iterating the above display, one gets the desired result.

(3) The dimension of $R$ is $2t-2$, hence $\dim \mathbb{F}[x_1,\ \dots,\ x_{2t-2}]=2t-2$.
\end{proof}

\begin{lemma}
\label{lemma:cm}
For each $k$ with $1\leqslant k\leqslant n-t+2$, the ring $R/\mathfrak{p}^{\langle k\rangle}$ is Cohen-Macaulay. Hence the ideal $\mathfrak{p}^{\langle k\rangle}$ is a maximal Cohen-Macaulay $R$-module; in particular, it is reflexive.
\end{lemma}

\begin{proof}
Since $R/\mathfrak{p}$ is a polynomial extension of a $(t-1)\times n$ Hankel determinantal ring, its multiplicity is
\[
e(R/\mathfrak{p})\ =\ \binom{n}{t-2}.
\]
Fix $k$ with $1\leqslant k\leqslant n-t+2$. Since $\Delta \in\mathfrak{p}^{\langle k\rangle}$, it follows from Lemma~\ref{lemma:watanabe} that $\mathfrak{p}^{\langle k\rangle}$ has radical~$\mathfrak{p}$. The associativity formula for multiplicities, \cite[Corollary~4.7.8]{Bruns-Herzog}, then gives the first equality in
\[
e(R/\mathfrak{p}^{\langle k\rangle})\ =\ \ell\left(\frac{R_\mathfrak{p}}{\mathfrak{p}^{\langle k\rangle}R_\mathfrak{p}}\right)e(R/\mathfrak{p})\ =\ k\binom{n}{t-2},
\]
while the second equality follows from Lemma~\ref{lemma:valuation}.

Let $A$ be the polynomial ring $\mathbb{F}[x_1,\dots,x_{n+t-1}]$, and let $P_k$ be the inverse image of $\mathfrak{p}^{\langle k\rangle}$ under the canonical surjection $A\longrightarrow R$. The images of the indeterminates
\[
{\boldsymbol{x}}\colonequals x_1, \dots, x_{t-2}, x_{n+1}, \dots, x_{n+t-1}
\]
are a homogeneous system of parameters for~$A/P_k = R/\mathfrak{p}^{\langle k\rangle}$. Set
\[
J\colonequals P_k+({\boldsymbol{x}})A.
\]
Using, for example, \cite[Corollary~4.7.11]{Bruns-Herzog}, one has
\begin{equation}
\label{equation:inequalities}
\ell(A/J)\ \geqslant\ e\big({\boldsymbol{x}},R/\mathfrak{p}^{\langle k\rangle}\big)\ \geqslant\ e\big(R/\mathfrak{p}^{\langle k\rangle}\big)\ =\ k\binom{n}{t-2}.
\end{equation}
We claim that
\[
\ell(A/J)\ \leqslant\ k\binom{n}{t-2}.
\]
Assuming the claim, all the terms in~\eqref{equation:inequalities} are equal, but then $R/\mathfrak{p}^{\langle k\rangle}$ is Cohen-Macaulay using, again, \cite[Corollary~4.7.11]{Bruns-Herzog}.

To prove the claim, consider the degrevlex order on $A$ induced by
\[
x_1>x_2>\cdots>x_{n+t-1}.
\]
Then the initial ideal of $J$ contains the ideal
\[
({\boldsymbol{x}})\ +\ (x_{t-1}, \dots, x_{n-k+1})^{t-1}\ +\ (x_t, \dots, x_n)^t,
\]
so it suffices to verify that the length of
\[
\frac{\mathbb{F}[x_{t-1},\dots,x_n]}{(x_{t-1}, \dots, x_{n-k+1})^{t-1} + (x_t, \dots, x_n)^t}
\]
is at most
\[
k\binom{n}{t-2}.
\]
This is immediate from the following lemma.
\end{proof}

\begin{lemma}
Let $\mathbb{F}$ be a field, and consider integers $t\geqslant 2$ and $1\leqslant r\leqslant s$. Then
\[
\ell\left(\frac{\mathbb{F}[y_1,\dots,y_s]}{(y_1, \dots, y_r)^{t-1} + (y_2, \dots, y_s)^t}\right)
\ =\ (s-r+1)\binom{s+t-2}{t-2}.
\]
\end{lemma}

\begin{proof}
When $r=1$, the length in question is that of
\[
\frac{\mathbb{F}[y_1]}{(y_1^{t-1})}\otimes_{\mathbb{F}}\frac{\mathbb{F}[y_2,\dots,y_s]}{(y_2, \dots, y_s)^t},
\]
which equals
\[
(t-1)\binom{s-1+t-1}{t-1}\ =\ s\binom{s+t-2}{t-2},
\]
so the asserted formula holds. Assume for the rest that $r\geqslant 2$.

The case when $t=2$ is readily checked as well; we proceed by induction on $t$ and $s$. Set
\[
V\colonequals\mathbb{F}[y_1,\dots,y_s]
\quad\text{and}\quad
I\colonequals (y_1, \dots, y_r)^{t-1} + (y_2, \dots, y_s)^t,
\]
and consider the exact sequence
\[
0\longrightarrow V/(I:y_2) \longrightarrow V/I \longrightarrow V/(I+y_2V) \longrightarrow 0.
\]
Since $(I:y_2) = (y_1, \dots, y_r)^{t-2} + (y_2, \dots, y_s)^{t-1}$, the inductive hypothesis gives
\begin{align*}
\ell(V/I) &\ =\ \ell(V/(I:y_2)) + \ell(V/(I+y_2V)) \\
&\ =\ (s-r+1)\binom{s+(t-1)-2}{(t-1)-2} + ((s-1)-(r-1)+1)\binom{(s-1)+t-2}{t-2} \\
&\ =\ (s-r+1)\binom{s+t-2}{t-2}.\qedhere
\end{align*}
\end{proof}

\begin{proof}[Proof of Theorem~\ref{theorem:class:group}]
By Lemma~\ref{lemma:watanabe}, the ring $R_\Delta$ is a localization of a polynomial ring, and hence a UFD. Nagata's theorem, e.g.,~\cite[page~315]{Bruns-Herzog}, then implies that $\operatorname{Cl}(R)$ is generated by the height $1$ prime ideals of $R$ that contain $\Delta$, namely by the ideal $\mathfrak{p}$.

Fix $k$ with $1\leqslant k\leqslant n-t+2$. Then $\mathfrak{p}^{\langle k\rangle}$ has radical $\mathfrak{p}$, and is unmixed by Lemma~\ref{lemma:cm}. Thus, the primary decomposition of $\mathfrak{p}^{\langle k\rangle}$ has the form $\mathfrak{p}^{(i)}$ for some $i$. The integer~$i$ can be computed after localization at $\mathfrak{p}$, but then Lemma~\ref{lemma:valuation} implies that $\mathfrak{p}^{\langle k\rangle}=\mathfrak{p}^{(k)}$ as claimed. Note that the ideal $\mathfrak{p}^{\langle k\rangle}$ is principal precisely when $k=n-t+2$. Lastly, each $\mathfrak{p}^{\langle k\rangle}$ is a maximal Cohen-Macaulay module by Lemma~\ref{lemma:cm}.
\end{proof}

\begin{remark}
\label{remark:mcm}
Let $Y$ be a~$t\times n$ matrix of indeterminates over a field $\mathbb{F}$, and consider the generic determinantal ring
\[
B\colonequals\mathbb{F}[Y]/I_t(Y).
\]
Set $P$ to be the prime ideal of $B$ generated by the size $t-1$ minors of the first $t-1$ rows of~$Y$, and $Q$ to be the prime generated by the size $t-1$ minors of the first $t-1$ columns. By \cite[Example~9.27(d)]{Bruns-Vetter}, the following are maximal Cohen-Macaulay $B$-modules:
\[
B,\ P,\ Q,\ Q^2,\ \dots,\ Q^{n-t+1},
\]
and, in fact, the only rank one maximal Cohen-Macaulay $B$-modules up to isomorphism. The canonical module of $B$ is isomorphic to $Q^{n-t}$, see \cite[Theorem~8.8]{Bruns-Vetter}.

Since the Hankel determinantal ring $R$ may be obtained as the specialization of $B$ modulo a regular sequence, it follows that the images in $R$ of the modules displayed above are Cohen-Macaulay $R$-modules. Note that~$\mathfrak{p}=PR$, and set~$\mathfrak{q}\colonequals QR$, in which case
\[
R,\ \mathfrak{p},\ \mathfrak{q},\ \mathfrak{q}^2,\ \dots,\ \mathfrak{q}^{n-t+1}
\]
are Cohen-Macaulay $R$-modules. Due to the symmetry in a Hankel matrix, one has
\[
\mathfrak{q}\ =\ \mathfrak{p}^{\langle n-t+1\rangle}\ =\ \mathfrak{p}^{(n-t+1)}.
\]
Fix $i$ with $1\leqslant i\leqslant n-t+1$. Since $\mathfrak{q}^i$ is a maximal Cohen-Macaulay $R$-module, and hence a divisorial ideal, it follows that
\[
\mathfrak{q}^i\ =\ (\mathfrak{p}^{(n-t+1)})^i\ =\ \mathfrak{p}^{(i(n-t+1))}\ \cong\ \mathfrak{p}^{(n-t+2-i)}.
\]
In particular, $\mathfrak{q}^{n-t+1}\cong\mathfrak{p}$, and the $n-t+3$ rank one maximal Cohen-Macaulay $B$-modules specialize to the $n-t+2$ elements of the divisor class group of $R$.

The canonical module $Q^{n-t}$ of $B$ specializes to the canonical module
\[
\mathfrak{q}^{n-t}\ \cong\ \mathfrak{p}^{(2)}
\]
of $R$. Since the $a$-invariant of the ring $R$ is $1-t$, and $\mathfrak{p}^{(2)}$ is generated in degree $t-1$, it follows that the \emph{graded} canonical module of $R$ is
\[
\omega_R\colonequals\mathfrak{p}^{(2)}.
\]
Note that the number of generators of $\omega_R$ as an $R$-module is
\[
\binom{n-1}{t-1}.
\]
Since $\omega_R$ is a reflexive $R$-module of rank one, it corresponds to an element $[\omega_R]$ of $\operatorname{Cl}(R)$. The order of this element is
\[
\operatorname{ord}\,[\omega_R]\ =\ \begin{cases} n-t+2 & \text{ if $n-t+2$ is odd},\\ (n-t+2)/2 & \text{ if $n-t+2$ is even}. \end{cases}
\]
\end{remark}

\section{\texorpdfstring{$F$}{F}-purity and the \texorpdfstring{$F$}{F}-pure threshold}

Following \cite[page~121]{Hochster-Roberts:purity}, a ring $R$ of positive prime characteristic is \emph{$F$-pure} if the Frobenius endomorphism $F\colon R\longrightarrow R$ is pure. We prove:

\begin{theorem}
\label{theorem:fpure}
Let $R$ be a Hankel determinantal ring over a field $\mathbb{F}$. If $\mathbb{F}$ has positive characteristic, then the ring $R$ is $F$-pure. If $\mathbb{F}$ has characteristic zero, then $R$ has log canonical singularities.
\end{theorem}

The proof uses the graded version of Fedder's criterion, \cite[Theorem~1.12]{Fedder}, and a result from \cite{Conca}; we record these below:

\begin{theorem}[Fedder's criterion]
\label{theorem:fedder}
Let $A$ be an $\mathbb{N}$-graded polynomial ring, where $A_0$ is a field of characteristic $p>0$. Let $I$ be a homogeneous ideal of~$A$, and set $R \colonequals A/I$. Let $\mathfrak{m}$ be the homogeneous maximal ideal of~$A$. Then $R$ is $F$-pure if and only if
\[
(I^{[p]}:_AI)\ \nsubseteq\ \mathfrak{m}^{[p]}.
\]
\end{theorem}

The following lemma can be seen as a special case of \cite[Theorem~3.12]{Conca} that express the primary decomposition of a product of Hankel determinantal ideals in terms of symbolic powers and the so-called gamma functions. We present here a direct argument that is based only on \cite[Lemma~3.7]{Conca}.

\begin{lemma}
\label{lemma:symbolic}
Let $A\colonequals\mathbb{F}[x_1,\dots,x_{s+r-1}]$ be a polynomial ring over a field $\mathbb{F}$, and let $H$ be the~$r\times s$ Hankel matrix in the indeterminates $x_1,\dots,x_{s+r-1}$. Set $I\colonequals I_t(H)$, where $t$ is an integer with $1\leqslant t\leqslant\min\{r,s\}$. Let $d$ be a positive integer, and let $\delta_1,\dots,\delta_m$ be minors of~$H$ such that $m\leqslant d$ and $\sum_i\deg\delta_i\geqslant td$. Then
\[
\delta_1\cdots\delta_m\ \in\ I^d.
\]
\end{lemma}

\begin{proof} By adding factors of degree $0$ if needed, we may assume that $m=d$. For $u$ an integer, set~$I_u\colonequals I_u(H)$. If $\deg\delta_i\geqslant t$ for all $i=1,\dots,d$, then the assertion is obvious. If~$\deg \delta_i< t$ for some $i$, say $u=\deg \delta_1< t$, then, since $\sum_i\deg\delta_i\geqslant td$, there must be an index~$j$ such that $\deg\delta_j>t$, say $v=\deg\delta_2>t$. By \cite[Lemma 3.7]{Conca} one has
\[
I_uI_v\subseteq I_{u+1}I_{v-1}
\]
since~$u+1<v$. Hence we may replace $\delta_1\delta_2$ in the product with $\delta'_1\delta'_2$, where $\deg\delta'_1=u+1$ and $\deg \delta'_2=v-1$. Repeating the argument as needed, we obtain the desired assertion.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{theorem:fpure}]
The characteristic zero case follows from the positive characteristic assertion by \cite[Theorem~3.9]{Hara-Watanabe}; in view of this, let $\mathbb{F}$ be a field of characteristic $p>0$. Set $A\colonequals\mathbb{F}[x_1,\dots,x_{n+t-1}]$ and $I\colonequals I_t(H)$. By Fedder's criterion, it suffices to verify that
\[
(I^{[p]}:_AI)\ \nsubseteq\ \mathfrak{m}^{[p]},
\]
where $\mathfrak{m}$ is the homogeneous maximal ideal of $A$. We construct a polynomial $f$ with
\begin{equation}
\label{equation:symbolic}
f\ \in\ I^{(n-t+1)}
\end{equation}
such that, with respect to the lexicographic order $x_1>x_2> \cdots >x_{n+t-1}$, one has
\[
\operatorname{in_{lex}}(f)\ =\ x_1x_2\cdots x_{n+t-1}.
\]
Since the initial term of $f$ is squarefree, it follows that $f^{p-1}\notin \mathfrak{m}^{[p]}$. We claim that~\eqref{equation:symbolic} implies $f^{p-1} \in (I^{[p]}:_AI)$, i.e.,
\[
f^{p-1}I\ \subseteq\ I^{[p]}.
\]
By the flatness of the Frobenius endomorphism of $A$, the set of associated primes of $A/I^{[p]}$ equals that of~$A/I$, so it suffices to verify that the containment displayed above holds after localization at the prime ideal $I$. The ideal $I$ has height $n-t+1$, so $(A_I,IA_I)$ is a regular local ring of dimension $n-t+1$, and the pigeonhole principle gives
\[
I^{(n-t+1)(p-1)+1}A_I\ \subseteq\ I^{[p]}A_I.
\]
Using~\eqref{equation:symbolic}, it follows that
\[
f^{p-1}IA_I\ \subseteq\ I^{(n-t+1)(p-1)+1}A_I,
\]
which proves the claim. It remains to construct $f$ with the properties asserted above; the construction depends on whether $n+t-1$ is odd or even:

Suppose $n+t-1$ is odd, set $k\colonequals (n+t)/2$. Then $I$ also equals the ideal generated by the size $t$ minors of the~$k \times k$ Hankel matrix
\[
H'\colonequals\begin{pmatrix}
x_1 & x_2 & x_3& \cdots & x_k\\
x_2 & x_3 & x_4 & \cdots & x_{k+1}\\
\vdots & \vdots & \vdots & \vdots & \vdots\\
x_k & x_{k+1} &\cdots & \cdots & x_{n+t-1}
\end{pmatrix}.
\]
Let $f$ be the product of $\delta_1\colonequals{[1\ \dots\ k \mid 1\ \dots\ k]}_{H'}$ and $\delta_2\colonequals{[1\ \dots\ k-1 \mid 2\ \dots\ k]}_{H'}$. Then
\[
\operatorname{in_{lex}}(f)\ =\ (x_1x_3 \cdots x_{n+t-1})(x_2x_4\cdots x_{n+t-2})\ =\ x_1x_2\cdots x_{n+t-1}
\]
as claimed. Let $\delta_3,\dots,\delta_{n-t+1}$ be size $t-1$ minors of $H'$. Then Lemma~\ref{lemma:symbolic} implies that
\[
\delta_1\cdots\delta_{n-t+1}\ \in\ I^{n-t+1}.
\]
Since $I$ is a prime ideal generated in degree $t$, and each of $\delta_3,\dots,\delta_{n-t+1}$ has degree $t-1$, it follows that $f=\delta_1\delta_2$ belongs to the symbolic power $I^{(n-t+1)}$, as claimed in~\eqref{equation:symbolic}.

When $n+t-1$ is even, set $k\colonequals (n+t-1)/2$, and consider the~$k \times (k+1)$ Hankel matrix
\[
H''\colonequals\begin{pmatrix}
x_1 & x_2 & \cdots & x_k & x_{k+1}\\
x_2 & x_3 & \cdots & x_{k+1} & x_{k+2}\\
\vdots & \vdots & \vdots & \vdots & \vdots\\
x_k & \cdots & \cdots & x_{n+t-2} & x_{n+t-1}
\end{pmatrix}.
\]
Then $I$ equals $I_t(H'')$. Take $f$ to be the product of the minors $\delta_1\colonequals{[1\ \dots\ k \mid 1\ \dots\ k]}_{H''}$ and~$\delta_2\colonequals{[1\ \dots\ k \mid 2\ \dots\ k+1]}_{H''}$, in which case
\[
\operatorname{in_{lex}}(f)\ =\ (x_1x_3 \cdots x_{n+t-2})(x_2x_4\cdots x_{n+t-1})\ =\ x_1x_2\cdots x_{n+t-1}.
\]
Choosing size $t-1$ minors $\delta_3,\dots,\delta_{n-t+1}$ of $H''$, Lemma~\ref{lemma:symbolic} gives
\[
\delta_1\cdots\delta_{n-t+1}\ \in\ I^{n-t+1}
\]
and hence $f\in I^{(n-t+1)}$, as in the previous case.
\end{proof}

The definition of $F$-pure thresholds is due to Takagi and Watanabe \cite{Takagi-Watanabe}, and provides a positive characteristic analogue of the log canonical threshold. We focus here on the $F$-pure threshold of a homogeneous ideal in standard graded $F$-pure ring:

\begin{definition}
Let $A$ be a polynomial ring over an $F$-finite field of characteristic~$p>0$, and let $I$ be a homogeneous ideal such that $R \colonequals A/I$ is $F$-pure. Let $\mathfrak{a}$ be a homogeneous ideal of~$R$, and let $J$ be its preimage in $A$. Given $e \in \mathbb{N}$, set
\[
\nu_e(\mathfrak{a})\colonequals\max\left\{r \geqslant 0 \mid (I^{[q]}:_AI)J^r \not\subseteq \mathfrak{m}_A^{[q]} \right\},
\]
where $q=p^e$. Then the \emph{$F$-pure threshold} of $\mathfrak{a}\subset R$ is
\[
\operatorname{fpt}(\mathfrak{a}) \colonequals \lim_{e \longrightarrow \infty}\nu_e(\mathfrak{a})/p^e.
\]
\end{definition}

Suppose, in addition, that $R$ is normal; let $\omega_R$ be the graded canonical module of $R$. Taking $\mathfrak{a}$ to be $\mathfrak{m}_R$ in the above definition, \cite[Theorem~4.1]{STV} implies that $-\nu_e(\mathfrak{m}_R)$ equals the degree of a minimal generator of $\omega_R^{(1-q)}$. Using this, we obtain:

\begin{theorem}
\label{theorem:fpt1}
Let $R=\mathbb{F}[x_1,\dots,x_{n+t-1}]/I_t(H)$, where $\mathbb{F}$ is a field of characteristic $p>0$, and $H$ is a $t\times n$ Hankel matrix. Then the $F$-pure threshold of $\mathfrak{m}_R\subset R$ is
\[
\operatorname{fpt}(\mathfrak{m}_R)\ =\ \frac{2(t-1)}{n-t+2}.
\]
\end{theorem}

\begin{proof}
Recall from Remark~\ref{remark:mcm} that $\omega_R=\mathfrak{p}^{(2)}$. For an integer $q=p^e$, one then has
\[
\omega_R^{(1-q)}\ =\ \mathfrak{p}^{(2(1-q))}.
\]
Write
\[
2(q-1)=i(n-t+2)+j,
\quad\text{ where }\ 0\leqslant j\leqslant n-t+1.
\]
In view of the graded isomorphism
\[
\mathfrak{p}^{(n-t+2)}\ \cong\ R(-(t-1)),
\]
one then has
\[
\omega_R^{(1-q)}\ =\ \mathfrak{p}^{(2(1-q))}\ =\ \mathfrak{p}^{(-i(n-t+2))}\mathfrak{p}^{(-j)}\ \cong\ \mathfrak{p}^{(-j)}(i(t-1)).
\]
Since $0\leqslant j\leqslant n-t+1$, the module $\mathfrak{p}^{(-j)}$ has minimal generators in degree $0$, which then implies that $\omega_R^{(1-q)}$ has minimal generators in degree $-i(t-1)$, and hence that
\[
\nu_e(\mathfrak{m}_R)\ =\ i(t-1)\ =\ (t-1) \left\lfloor\frac{2(q-1)}{n-t+2}\right\rfloor.
\]
The calculation of $\operatorname{fpt}(\mathfrak{m}_R)$ follows immediately from this.
\end{proof}

Using the general theory developed in \cite{Henriques-Varbaro}, one can also compute the $F$-pure threshold of the ideal $I_t(H)$ in the polynomial ring $\mathbb{F}[x_1,\dots,x_{n+t-1}]$:

\begin{theorem}
\label{theorem:fpt2}
Let $H$ be a $t\times n$ Hankel matrix of indeterminates over a field $\mathbb{F}$ of positive characteristic. Then the $F$-pure threshold of $I_t(H)\subset \mathbb{F}[x_1,\dots,x_{n+t-1}]$ is
\[
\operatorname{fpt}(I_t(H))\ =\ \min\left\{\frac{n+t-2i+1}{t-i+1}\mid i=1,\dots,t\right\}.
\]
More precisely, if $\lambda\in \mathbb{R}_{>0}$, the generalized test ideal $\tau(\lambda \bullet I_t(H))$ is
\[
\tau(\lambda \bullet I_t(H))\ =\ \bigcap_{i=1}^tI_i(H)^{(\lfloor\lambda(t-i+1)\rfloor-n-t+2i)}.
\]
\end{theorem}

\begin{proof}
The powers of the ideal $I_t(H)$ are integrally closed by \cite[Theorem~4.5]{Conca}, and using~\cite[Theorem~3.12]{Conca} one has
\[
\bigcup_{s\geqslant 1}\operatorname{Ass} I_t(H)^s\ \subseteq\ \{I_1(H),\ I_2(H),\ \dots,\ I_t(H)\}.
\]
In the notation of \cite[\S\,3]{Henriques-Varbaro}, by \cite[Theorem~3.12]{Conca} we also infer that $I_t(H)$ satisfies condition ($\diamond$) and that
\[
e_{I_i(H)}(I_t(H))\ =\ t-i+1
\quad\text{ for }\ i=1,\dots,t.
\]

Recall that the polynomial $f$ constructed in the proof of Theorem~\ref{theorem:fpure} has a squarefree initial term. By an argument similar to the one used there for $i=t$, one sees that
\[
f\ \in \ I_i(H)^{(n+t-2i+1)}
\quad\text{ for }\ i=1,\dots,t.
\]
Since $I_i(H)$ is equal to the ideal generated by the size $i$ minors of an $i\times (t+n-i)$ Hankel matrix, it follows that
$\operatorname{height}\, I_i(H)=n+t-2i+1$, and that the ideal $I_t(H)$ satisfies the condition ($\diamond+$). The assertion now follows by \cite[Theorem~3.14]{Henriques-Varbaro}.
\end{proof}

\begin{remark}
A similar argument allows one to compute the $F$-pure threshold and the generalized test ideals (in positive characteristic), as well as the log canonical threshold and the multiplier ideals (in characteristic zero), of any product of ideals of minors of a Hankel matrix in a polynomial ring.
\end{remark}

We conclude with the following question; we prove in \cite{CMSV} that the answer is affirmative in a number of cases.

\begin{question}
Is every Hankel determinantal ring over a field of positive characteristic an~$F$-regular ring?
\end{question}

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}
Ultra-luminous X-ray sources (ULXs) are variable, off-nuclear X-ray
sources in nearby galaxies, with luminosities in excess of $L \simeq
10^{39}~ {\rm erg}~{\rm s}^{-1}$ (the Eddington luminosity of an $M =
10~ M_{\odot}$ black hole).  Though they are luminous, most ULXs are
likely familiar objects.  The well-known source GRS 1915$+$105, for
instance, can by modestly super-Eddington for its mass, depending on
the model and energy band considered.  The vast majority of ULXs only
exceed $L \simeq 10^{39}~ {\rm erg}~{\rm s}^{-1}$ by a small margin
(Swartz et al.\ 2011; for a recent review, see Feng \& Soria 2011).

The small subset of ULXs with luminosities of $L \simeq 10^{40}~ {\rm
  erg}~ {\rm s}^{-1}$ and above are potentially more interesting, as
they might be powered by accretion onto so-called
``intermediate-mass'' black holes (IMBHs), or an accretion flow that
has genuinely defeated the isotropic Eddington limit.  Of course, this
small sub-class could be a combination of these two phenomena.  An
alternative possibility is that the emission from such sources is
actually anisotropic, perhaps owing to a ``funnel'' in the inner
accretion disk (King et al.\ 2001).

{\it XMM-Newton} has revolutionized studies of ULXs, making it
possible to obtain spectra that require multiple components.  Early
efforts to decompose the best ULX spectra found evidence of separate
soft and hard components.  The soft components could be fit with disk
models, and low temperatures obtained -- generally $kT =$0.2--0.3~keV
-- provided some evidence of accretion onto IMBHs since $T \propto
M_{BH}^{-1/4}$ and $kT = 1$~keV is typical for accretion onto
stellar-mass black holes close to the Eddington limit (e.g. Miller et
al.\ 2003; Miller, Fabian, \& Miller 2004).  This spectral
decomposition is based on a close analogy with better-known X-ray
binaries and may not be unique nor appropriate for ULXs.  However,
subsequent studies of soft component variability found that these
would-be disks may not be blackbody-like (Kajava \& Poutanen 2009;
Feng \& Kaaret 2007, 2009; also see Soria 2007).  Other recent work
has discovered a spectral roll-over above the Fe K band (Gladstone,
Roberts, \& Done 2009).  This can be interpreted as evidence of
super-Eddington accretion, though the spectra still require
independent soft components with low characteristic temperatures.  In
this work, we examine the nature of the best ULX spectra from nearby
sources with $L \simeq 10^{40}~ {\rm erg}~ {\rm s}^{-1}$, factoring in
recent studies that impact the possibility of winds and variable
absorption.

\section{Sample and Data Reduction}
We considered the sample of nearby $L \simeq 10^{40}$~erg/s sources
observed with {\it XMM-Newton} and reported in Miller, Fabian, \&
Miller (2004), Gladstone, Roberts, \& Done (2009), and Kajava \&
Poutanen (2009).  ULXs at distances greater than 5~Mpc were not
considered owing to their reduced photon flux.  Of the remaining
sources, NGC 1313 X-2 was not considered as any evidence that it
exceeds $L = 10^{40}$~erg/s appears weak and highly model--dependent,
and evidence for a cool disk component is only marginal (e.g. Miller
et al.\ 2003).  IC 342 X-1 -- which is near to $L = 10^{40}$~erg/s --
was excluded owing to the modest significance of a putative cool
component.  In exploratory fits, the column density along the line of
sight to IC 342 X-1 fell in the $N_{H} = 0.5-1.5\times 10^{22}~ {\rm
  cm}^{-2}$ range, greatly complicating the detection of $kT \simeq
0.2$~keV emission.  With these constraints, our sample includes
Holmberg IX X-1, NGC 1313 X-1, Ho II X-1, NGC 5408 X-1, and NGC 5204
X-1.  All archival {\it XMM-Newton} observations of these sources were
downloaded and reduced using SAS version 12.0.1.

The EPIC-pn camera has the highest collecting area across the full
0.3--10.0~keV band, and is best calibrated for spectral fitting.  For
simplicity and self-consistency, our analysis was restricted to
spectra obtained using the EPIC-pn camera.  Unlike {\it Chandra}, {\it
  XMM-Newton} does not dither, and when a source lands within the gap
between chips in EPIC cameras, its effective exposure and encircled
energy fraction can be affected, ultimately compromising flux
estimates.  Observations wherein the source image fell within chip
gaps were therefore rejected, in order to ensure robust flux
determinations.

Background regions were extracted on the same CCD as the source, and
generally using an extraction region of the same size.  The background
regions were analyzed to identify periods of background flaring, and
to create a GTI file to exclude these intervals when extracting events
for spectral analysis.  Spectra, backgrounds, and responses were then
calculated using the appropriate tools.  All spectra were grouped to
require at least 25 counts per bin using the FTOOL ``grppha'', in
order to ensure the validity of $\chi^{2}$ statistical tests.

\section{Analysis and Results}
All spectra were fit using XSPEC version 12.2 (Arnaud et al.\ 1996).
Neutral interstellar absorption was fit using the ``tbabs'' model.  As
required by ``tbabs'', the ``vern'' atomic cross-sections and ``wilm''
elemental abundances were used.  The Milky Way's contributions to the
total neutral column density along these lines of sight are small, and
we therefore used a single ``tbabs'' component to account for both
Galactic absorption and the column within each ULX host galaxy.  Solar
abundances were assumed in all fits (but see the Discussion).

The spectral resolution afforded by dispersive spectrometers has
recently been leveraged to address the extent to which absorption may
drive spectral evolution in accreting systems (Miller, Cackett, \&
Reis 2009).  The depth of individual photoelectric absorption edges
remains remarkably constant across a broad range in luminosity in
low-mass X-ray binaries, in binaries with potential
``intermediate-mass'' stars such as Cygnus X-2, and even in Cygnus X-1
(which accretes from an O 9.7 Iab supergiant).  This argues that the
line of sight column density should be held constant in spectral fits.
Current limits on Fe K emission and absorption lines in ULXs are
commensurate with detections in Galactic X-ray binaries, and far below
expections if line strengths scale with the mass accretion rate
(Walton et al.\ 2012, 2013).  We therefore fit all EPIC-pn spectra of
a given ULX jointly, such that the interstellar column density was
jointly determined and had a common value for every spectrum.

For simplicity, we chose to fit the soft, potentially thermal
components with the well-known ``diskbb'' model (Mitsuda et
al.\ 1984).  To characterize the hard flux in each ULX spectrum, we
used the ``compTT'' model (Titarchuk 1994).  The use of ``compTT'' is
important for characterizing the turn--over seen in the 6--10~keV band
in many sensitive spectra of ULXs with $L \simeq 10^{40}$~erg/s
(e.g. Gladstone et al.\ 2009; Walton et al.\ 2013).  It produced
statistically superior fits to simple power-law models for the hard
flux.  The temperature of the low-energy thermal distribution $T_0$
was linked to that of the disk component.  The other crucial parameters
within ``compTT'', the electron temperature $kT_e$ and the optical
depth $\tau$ were allowed to vary.  However, few spectra are able to
constrain both parameters, and in those cases a value of $kT_e = 2.0$~keV
was adopted.  This value is broadly consistent with the spectra where
constraints were possible, and also broadly consistent with the values
reported in fits with ``compTT'' reported by Gladstone et al.\ (2009)
and Feng \& Kaaret (2009).

An additional diffuse emission component is present in the spectra of
NGC 5408 X-1, likely due to local warm gas and star formation.
Emission localized around 1 keV can be modeled with a ``mekal'' plasma
with $kT = 0.87(2)$~keV and a normalization of $8.0\pm0.5 \times
10^{-5}$.  These values were determined through joint fits.  This
correctly accounts for diffuse emission with a constant flux.

The results of our spectral fits are given in Table 1.  The procedure
of jointly determining the column density is clearly one that allows
for excellent fits.  In all cases, the joint fit returns a reduced
$\chi^{2}$ statistic that is close to unity.  The use of the
``compTT'' component with a turn-over above the Fe K band also
contributed to the excellent fits.  Figure 1 plots the X-ray
luminosity measured in the putative disk components, versus their
color temperature values.  Both in Table 1 and Figure 1, the
luminosity is restricted to the band in which the flux was actually
observed (0.3--10.0~keV).

Figure 1 also plots the relationship expected for simple blackbody
emission, with a number of different normalizations.  It is
immediately apparent that the ULX putative disk components show a
clear, positive relationship between luminosity and temperature that
is qualitatively consistent with $L \propto T^{4}$.  (Note that band
pass effects can be important in effective luminosity versus
temperature relationships; it is possible that $L \propto T^{5}$ may be
anticipated for very cool disks measured in the 0.3--10.0~keV band.)
The lone exception is NGC 5408 X-1; the points from that source are
tightly clustered.

In order to quantify the apparent relationships, we ran statistical
correlation tests on the data from each individual source, and also
made simple least-squares fits to determine the slope of the data.
The results of these tests and fits are listed in Table 2.  Errors on
both luminosity and temperature were considered in fitting the data
and estimating errors on the slope.  Strong positive correlations are
confirmed in each source, apart from NGC 5408 X-1.  The data from Ho
IX X-1 and Ho II X-1 are formally consistent with $L \propto T^{4}$,
and the slope of NGC 1313 X-1 is consistent within 1.5$\sigma$.  The
slope of NGC 5204 X-1 is somewhat flatter.  The best traces of $L$ versus
$kT$ in stellar-mass black hole disks -- made using {\it Swift},
spanning three orders of magnitude in $L$, and also obtained using
``diskbb'' and ``compTT'' continuum components -- find more shallow
slopes, e.g. $L \propto T^{3.3\pm 0.1}$ (Rykoff et al.\ 2007, also see
Reynolds \& Miller 2013, Salvesen et al.\ 2013).

The errors on the slopes listed in Table 2 are large owing to the small
number of points available.  Similarly, the strongest correlations are
only significant at the 99\% level of confidence.  However, taken
literally, the data would suggest that the putative cool thermal disk
components in this small subset of extreme ULXs may indeed represent
disk emission.  Since disk tempertures this low are only seen in
stellar-mass black holes at or below $0.01~L_{Edd.}$, the results
would nominally indicate accretion onto more massive black holes,
since $T \propto M_{BH}^{-1/4}$ for black hole accretion.

\section{Discussion and Conclusions}
We have examined the relationship between the color temperature of
putative cool, thermal components versus their luminosity, in the
0.3--10.0~keV spectra of ULXs with $L_{tot} \simeq 10^{40}$~erg/s.  We
find evidence of positive correlations between the luminosity of these
putative disk components, and their color temperature.  In the
observed band, some temperature and luminosity trends are formally
consistent with the $L \propto T^{4}$ relationship expected for simple
blackbody emission from a standard thin accretion disk.  In all but
one source, the data are consistent with a slightly flatter
relationship that is observed in stellar-mass black holes in the same
band (e.g. $L \propto T^{3.3\pm 0.1}$, Rykoff et al.\ 2007; also see
Reynolds \& Miller 2013, Salvesen et al.\ 2013).  The lone exception
is NGC 5408 X-1, for which the data span a very small range in
luminosity and temperature, and thus no trend can be discerned.  Taken
literally, these results may support an interpretation of these
components as emission from cool accretion disks around
intermediate-mass black holes.  We therefore proceed to make a
critical examination of our methods and assumptions in this section.

As noted previously, some recent work has found putative disk
luminosity and temperature to be anti--correlated when fitting ULX
spectra with disk components at low energy (e.g. Feng \& Kaaret 2007,
2009; Kajava \& Poutanen 2009).  Our analysis employed a more
restrictive selection criterion, in that observations wherein the
source image landed on a chip gap were excluded.  Observations with
spurious or uncertain flux measurements are therefore omitted.
Moreover, the correlation we have found is specific: the {\it disk}
luminosity and the {\it disk} temperature are positively correlated.
We do not find strong correlations between {\it total} luminosity and
disk temperature.  Though coronae and disks must be linked, the need
for magnetic processes (see, e.g., Merloni \& Fabian 2002) means that
the luminosity of these components can be decoupled at times.

Our analysis also differs from prior efforts in that the column
density along the line of sight to a given source was jointly
determined by the numerous spectra, and not allowed to vary between
them.  It is possible that this method is faulty, especially
if ULXs are fueled by massive stars with variable, clumpy winds.
However, existing data appear to justify our assumptions and fitting
methods:

First, grating spectra of even Cygnus X-1 find a consistent column
across the binary phase, except perhaps when the O star is closest to
our line of sight (Miller, Cackett, \& Reis 2009).  Second, existing
spectra of NGC 1313 X-1 and Holberg IX X-1 now place extremely
restrictive limits on emission and absorption features in these
sources (Walton et al.\ 2013).  Any emission lines must have lower
equivalent widths than the lines seen in Galactic X-ray binaries with
massive companions.  This indicates that companion winds are largely
absent or very highly ionized.  Either way, companion winds are
unlikely to contribute to an evolving neutral column density.  Limits
on absorption lines are also below those detected from disk winds in
stellar-mass black holes and many AGN, again limiting the scope for
variable absorption (Walton et al.\ 2012, 2013; Pasham \& Strohmayer
2013).  Last, where dips are detected in ULXs (e.g. NGC 5408 X-1,
Grise et al.\ 2013), they are apparently quasi--periodic (Pasham \&
Strohmayer 2013), and thus inconsistent with clumps in a companion
wind.  Dips in Cygnus X-1 are clustered at $\phi = 0.95$ and $\phi =
0.6$, with those at $\phi = 0.6$ likely due to the accretion stream
impacting the outer disk; dips owing to clumps in the companion wind
appear to be spread randomly in phase (Balucinksa-Church et
al.\ 2000).  And, as noted previously, even in Cygnus X-1, the optical
depth in various edges is remarkably constant (e.g. Miller, Cackett,
\& Reis 2009).

In our treatment of the neutral column, we also assumed solar
abundances for all elements.  Some studies of the dwarfs in this
sample (in particular) have reported sub--solar metallicity values in
the ISM of those galaxies (e.g. Guseva et al.\ 2011, Egorov,
Lozinskaya, \& Moiseev 2013), and caution is warranted.  Fits to the
best X-ray spectra, however -- including gratings spectra obtained
with the RGS -- find abundances consistent with solar (Winter,
Mushotzky, \& Reynolds 2007; Pintore \& Zampieri 2012).  The same
effect is found in our data: if the absorption model for Ho II X-1 in
Table 1 is altered so that the metal abundances are only 10\% of
solar, a significantly worse fit is achieved ($\Delta \chi^{2} = 44$).
The same effect holds for the large spiral NGC 1313 ($\Delta \chi^{2}
= 132$).  Such results are not driven by contributions from the
Galactic column; in all cases, the Galactic column (as estimated by
Dickey \& Lockman 1990) is only a fraction of that measured.
Moreover, it is important to remember a basic facet of the observed
absorption edge features: one can trade intrinsic column density and
abundance, but their product has to match the observed edge depth.
Shifting abundance values will affect all spectra from a given source
in the same manner.

The spectral model we employed also has difficulties.  The ``comptt''
model includes thermal emission, and it therefore competes with the
external ``diskbb'' component for the soft X-ray flux.  In this sense,
it is not perfectly self-consistent.  However, ``comptt'' is required
to describe the roll-over in the 5--10 keV band (e.g. Gladstone,
Roberts, \& Done 2009), and this is now a standard model.  Alternative
Comptonization treatments, such as ``simpl'', only produce a power-law
and thus miss the observed roll-over.  For the fits presented in Table
1, the additional soft component is required by the data at extremely
high statistical significance.

There are important caveats, but our results nominally support the
possibility that soft components in ULX spectra represent emission from
standard accretion disks extending to the ISCO.  It is possible to
derive some simple mass estimates by scaling from stellar-mass black
holes: $M_{ULX} \simeq (M_{XRB} / M_{\odot}) \times
(T_{XRB}/T_{ULX})^{4}$, where $T_{XRB}$ is the disk temperature
typical for stellar-mass black holes in X-ray binaries close to the
Eddington limit (or, in a state analogous to that in which ULXs
accrete).  If we take $kT_{XRB} = 1$~keV, $M_{XRB} = 10~ M_{\odot}$,
$T_{ULX} = 0.2$~keV, then a mass of $M_{ULX} = 6250~ M_{\odot}$ is
implied.  However, not all transients that cycle through each of the
canonical spectral states are observed to have maximum temperatures of
$kT \simeq 1$~keV.  XTE J1650$-$500, for instance, had a maximum
temperature of $kT \simeq 0.6$~keV (e.g. Reis et al.\ 2013).  For
$kT_{XRB} = 0.5$~keV, $M_{XRB} = 5~ M_{\odot}$, and $kT_{ULX} =
0.25$~keV, a more modest mass of $M_{ULX} \simeq 80 M_{\odot}$ is
implied.  Clearly, the mass estimate is strongly dependent upon the
``typical'' temperature of a standard X-ray binary disk at Eddington.

These mass estimates are simplified.  There is no certainty that ULXs
radiate at Eddington.  Moreover, the data allow for flatter
relationships between luminosity and temperature, potentially
suggestive of a changing disk radius.  In this circumstance, and in
situations where the coronal energy drains the disk, mass estimates
are more complex, and generally imply lower masses ($M_{ULX} \leq 100~
M_{\odot}$, Soria 2007).  The putative cool disk components in our
models only emit a fraction of their luminosity in the observed
(0.3--10.0~keV) band.  However, these disks would dominate the source
luminosity in the 0.01--10.0~keV band, and would then be more
analogous with the disk--dominated ``high/soft''states seen in
stellar-mass black holes.  Depending on numerous details, the slopes
measured when considering bolometric disk luminosity versus
temperature may be 0.5--1.0 flatter than the values reported in Table
2.  This effect is within the error ranges quoted in the existing
fits, and assumes that the spectral model can safely be extrapolated
to lower energy values.

It is possible that the observed temperatures and flux trends could
represent emission from a locally--inhomogeneous disk, although the
observed temperature contrast is slightly greater than that envisioned
in current treatments (e.g. Dexter \& Quataert 2012).  It is also
possible that the putative cool disk components we have studied do not
originate close to the black hole, but rather outside of some
transition radius, with an hot, inner, super-Eddington disk
represented by the Comptonized component.  In this scenario, however,
considerable fine--tuning would likely be required for the different
sources and transition radii to create the observed temperatures and
trends.  Future monitoring campaigns aimed at improved traces of soft
component variability, and very deep observations that search for the
outflows expected in super-Eddington regimes, may help to better
understand this subset of ULXs.