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US Supreme Court Center > Volume 419 > CIOFFI v. U.S 419 U.S. 917 (1974) > Full Text
CIOFFI v. U.S - 419 U.S. 917 (1974)
CIOFFI v. U.S. , 419 U.S. 917 (1974)
Salvatore CIOFFI
No. 73-1727.
On petition for writ of certiorari to the United States Court of Appeals for the Second Circuit.
The petition for a writ of certiorari is denied.
Mr. Justice DOUGLAS, with whom Mr. Justice BRENNAN concurs, dissenting.
Petitioner was convicted of obstruction of justice and conspiracy to obstruct justice by threatening, intimidating, or otherwise endeavoring to influence a grand jury witness. The witness, one Perry Scheer, had been a principal in a collapsed brokerage house whose activities were under investigation by the SEC and the FBI; there was strong evidence that Scheer had been involved in at least seven illegal securities transactions. Petitioner, who was allegedly in league with various persons who could have been harmed by Scheer's testimony before a grand jury investigating the affairs of the brokerage house, met with Scheer on several occasions and sought to secure Scheer's silence through veiled threats and suggestions that Scheer 'take the Fifth' (or, in more contemporary parlance, 'stonewall it'). Unbeknownst to petitioner, Scheer by this time was cooperating fully with federal authorities, and had been fitted out with a recording device on which he recorded several of his conversations with petitioner; these recordings were introduced at trial to corroborate and supplement Scheer's own testimony, and were played several times for the jury. I am unable to agree that the use of recordings made under
Page 419 U.S. 917 , 918
such circumstances is consistent with constitutional guarantees.
In a series of decisions beginning with On Lee v. United States, 343 U.S. 747, this Court has held that electronic interception or recording of conversations with the consent of one party does not violate Fourth Amendment standards. [Footnote 1] With one notable exception, however, these decisions were handed down prior to Katz v. United States, 389 U.S. 347, in which we formally interred much of their conceptual underpinning by holding that Fourth Amendment protections rest upon reasonable expectations of privacy rather than upon common-law property principles. That exception came in United States v. White, 401 U.S. 745, where four Members of the Court held that On Lee remained good law after Katz, four other Members maintained with equal adamance that it did not, and Mr. Justice Black (adhering to his dissent in Katz) found it unnecessary to reach the issue.
At a bare minimum, Katz must be read to require that monitoring of this sort be conducted only pursuant to a warrant: "Over and again this Court has emphasized that the mandate of the [Fourth] Amendment requires adherence to judicial processes,' United States v. Jeffers, 342 U.S. 48, 51, and that searches conducted outside the judicial process, without prior approval by judge or magistrate, are per se unreasonable under the Fourth Amendment-subject only to a few specifically established and well-delineated exceptions.' 389 U.S., at 357 (footnotes omitted). In the absence of such judicial supervision, there is no effective safeguard against the possibility of an uncontrolled electronic police state. [Footnote 2]
In my view the Executive Branch acts unlawfully when it invades an individual's privacy through trickery or fraud. See, e. g., Lewis v. United States, 385 U.S. 206, 344-347 ( Douglas, J., dissenting); Hoffa v. United States, 385 U.S. 293, 347-348 (separate opinion of Douglas, J.). The dangers posed by such invasions become particularly acute, however, when they are achieved through or accompanied by electronic monitoring of the sort presented here. [Footnote 3]
I would grant certiorari.
Footnote 1 See, e. g., Lopez v. United States, 373 U.S. 427; Osborn v. United States, 385 U.S. 323, 17 L. Ed.2d 394; United States v. White, 401 U.S. 745.
Footnote 2 See United States v. White, 401 U.S., at 760 ( Douglas, J., dissenting); id., at 755-756 (Brennan, J., concurring). In Osborn v. United States, 385 U.S. 323, 17 L. Ed.2d 394, the Court upheld the use of a recording of a face-to-face conversation where there had been prior dudicial authorization for the recording. Id., at 329-330. In my dissent from Osborn and elsewhere, I have set forth my view that even prior judicial approval cannot validate intrusions into constitutionally protected zones of privacy for the seizure of mere evidentiary material; the need for this protection is particularly acute when the items to be seized are an individual's own words, thoughts, papers and personal effects, even if no Fifth Amendment problem is squarely presented. Id., at 349-354 (Douglas, J., dissenting); Warden v. Hayden, 387 U.S. 294, 319-325 (Douglas, J., dissenting); Couch v. United States, 409 U.S. 322, 343 (Douglas, J., dissenting). In light of the absence of any attempt to secure judicial authorization in the present case, I see no need to press that view further at this time.
Footnote 3 See, e. g., Osborn v. United States, 385 U.S., at 352-354 (Douglas, J., dissenting); United States v. White, 401 U.S., at 762- 765 (Douglas, J., dissenting); Lopez v. United States 373 U.S., at 463-471 (Brennan, J., dissenting).
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Booze without the cruise: businesses offering to-your-door liquor delivery
Regina market ripe for liquor deliveries, says business owner
Brandon Harder • Regina Leader-Post
Feb 23, 2018 • Last Updated February 23, 2018 • 2 minute read
Caitlin Christoph, owner of the Booze Crew Courier Services Inc., puts a case of beer in her trunk at the Victoria Avenue East Liquor Store in Regina. Photo by TROY FLEECE /Regina Leader-Post
Liquor and cigarettes, delivered right to the doorstep.
That is, in a nutshell, the business model of Regina startup the Booze Crew, which hopes to begin deliveries mid-March.
While the service might keep the party rolling for some, owner Caitlin Christoph says she hopes it will keep them from getting behind the wheel.
"There's a lot of drunk drivers around. It's really close to heart," she said.
"I love my family and friends enough to be proactive."
In addition, the single mother hopes she can turn a healthy profit. When a similar business recently shut down Regina operations, she says she saw it as a window of opportunity.
"We thought, OK there's a need for it — to take on those customers who won't be getting serviced anymore."
Christoph's application to SLGA for the licence to deliver such products is still pending. However, if it is approved, the Booze Crew plans to operate with three drivers.
Cash and card options are to be available for payment. Customers will need to show proof they are over 19 and sign a receipt, which the company will be obligated to keep in its files for SLGA inspection.
Business hours will be Monday to Thursday from 9:30 a.m. to 9 p.m.
On Fridays and Saturdays hours will be extended until midnight.
While some customers might still be pining for another pint or two after the stroke of midnight, Christoph says she feels "risk factors" for her drivers would increase after that time.
"There's a lot more things that can happen. People are more likely to rob."
"It's happened to one of the companies here," she added.
As such, customers wanting service after midnight will have to dial a different business. They should have options, according to SLGA, which says there are actually 10 companies with home delivery licenses in Regina (38 in the province), not including the Booze Crew.
Among current options for late night delivery is Dial-A-Brew. And while the Booze Crew is planning to deliver liquor and cigarettes only, Dial-A-Brew has no such limitations.
"I deliver just about anything, as long as it's legal," owner Orrin Zannet said with a chuckle.
Zannet is also the company's sole deliveryman.
Regina is somewhat of an "untapped market," he said, noting that on a good night he can make more than forty deliveries. And Saskatchewan people are still learning about the service.
"I get a lot of customers who are shocked when I even answer the phone," he said.
Zannet does the lion's share of his business between 10 p.m. and 3 a.m.
He only accepts card or e-transfer payments, he said, noting that helps lessen the likelihood of having "a target on my back."
That said, some of his customers have limited mobility and the more mobile ones have never given him trouble.
"Knock on wood. I've had no issues."
In his experience, his customers would rather chat his ear off than fight.
bharder@postmedia.com
twitter.com/old_harder

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Diabetes is manageable with the right tools and support so don't be afraid to find out more, ask questions and ask for help when you need it. Whether you have diabetes or not you can either be that support or the one seeking and we can all #BeRemarkable. Remember that!
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Our oldest son is a cancer survivor and because of some of the treatments involved developed Diabetes 1 when he was 16. He just took it in stride as something else to adjust to and keeps going forward in his life never complaining about the issues he deals with.
my uncle Bob has diabetes and if the most hardworking caring man I know, yes he is my favorite uncle. He makes everyone feel special. He just turned 86 and his diabetes has never slowed him down, he is an excellent chef and I love to visit him.
My dad because when he found out he was diabetic he quit a 40 year smoking habit full stop!
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My uncle passed away from diabetes 17 years ago, he was remarkable because he never complained about his situation he just dealt with it.
My mother who is 85 lives in her own home. She has diabetes and arthritis. She has a leg that is basically a dead weight. She uses a walker and it take a lot out of her just to get in bed. Just wish I lived closer to help her out more.
My Uncle Lou traveled around the world for years. He developed diabetes and is now on dialysis. His illness has barely slowed him down. He still likes to travel when he has a chance.
My friends daughter has diabetes and she is so remarkably calm and responsible about managing it.
I know someone who has diabetes, very sadly, since the age of 5. This individual continues to be very dedicated hockey player into his teens and is extremely careful and wise regarding his own sugar levels!
My Grandmother was diabetic. She had numerous health problems but she was determined and never let them get the better of her – she lived until her 90s. She was an awesome Nana to all of us kids and we used to love going to visit at her house. She had 5 kids, 12 grandchildren and 10 great grandkids. We all loved and respected her immensely.
Diabetes runs in my Mom's family and my Uncles are doing great with keeping it under control and living their lives to the fullest.
My grandfather was diabetic. He taught me how to play guitar, was a hard worker and was just all around amazing.
it seems more and more people I know are being diagnosed. most recent my sister in law. she has made a lifetime eating habit change and looks and feel amazing she says.
My good friend's son has diabetes and he is so brave and patient. I totally admire his positive attitude and his dedication to staying healthy.
My mom has diabetes and she is so amazing! She is our rock!
A co-worker of mine has diabetes and she never misses any work and is always so upbeat.
I think my father in law is remarkable because had diabetes but it didn't stop him from doing what he loved, which was to fix cars.
My husband has recently been diagnosed with diabetes..after a severe car accident came on the diabetes..link of diabetes and high blood pressure and stress..you bet!
I think my niece is remarkable! She was just diagnosed with diabetes at the age of five. She is such a trooper and is a total pro at taking her own blood and testing her sugars.
My husband has diabetes and has now taken up running to help get in shape. He is doing so well. He is up to running 5k! I'm so proud of him.
A friend of ours has diabetes, manages it well, and hasnèt slowed down one bit.
I have a friend that was diagnosed with diabetes a couple yrs ago and there is so much along with it. She had to change everything.
My Grandma is remarkable. She has been through so much and always focuses on others instead of herself.
My husband has diabetes and went from pills to the needles. He is 69 and he doesn't let this get him down. He manages it very well.
I did all the steps on the POWERED BY ACCURACY SITE for fb and twitter but I cannot find the links on my pages, will try again tomorrow!
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My cousin has had diabetes for over 25 years, unfortunately he lost the lower part of his leg but it has never stopped him, he was a police officer and he still managed to pass his fitness test every year!
My manager has diabetes and I think he's remarkable. He works a lot, takes care of his two kids and is very careful to manage his diabetes the way he should. It's very impressive.


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/American History
Essay by review • March 1, 2011 • Essay • 930 Words (4 Pages) • 1,742 Views
Essay Preview: 1642
There were many important people in history that were born and living in the year 1642. Isaac Newton and Taylor Edward are two of the many important people of 1642. The Caroline age was also a part of the year 1642.
Isaac Newton was a very inspirational person of the year 1642. Many different kinds of people that out Isaac Newton to be not only one of the greatest figures in the history of mathematics but also on r of the smartest intellects of the human race. Beside his explanation of the laws of motion and gravitation and his discoveries concerning light and colornks development of calculus is regarded as the greatest creation in all of mathematics. His dispute with german mathematicion and philosopher Gohfried Leibniz (1616-1716) over calculus was one of the longest and bitterest scientific disputes. In the same year, Italian mathematician Galileo (1564-1642) died, Isaac Newton was born in the village of wools Thorne, near Lincolnshire, England. So small and sickly was the premature infant Newton that no one thought he would survive. He was so tiny that he could have fit in a quart jar. Newton's father, also named Isaac, died before he was born. When Newton's mother remarried three years later after his father's death, he went to live with hi maternal grandmother. Having never known his father and feeling angry sand jealous that his mother had another husband, Newton was often resentful and solitary while living with his grandmother. Newton was sent to Kings School in Grantham at age 12, but he spent most of his spare time building models and many devices of all sorts. He built kites with lanterns in them, a mill powered by a mouse, and water clocks, he also made sketches of plants, animals, and birds. At school he showed no special ability, but when pushed by the class bully, who was also first in the class, Newton managed to beat
him in academics and in an actual fight. When his mother took him out of school at 17, intending for him to work at home and become a farmer, Newton was uninterested, lazy, and distracted. His uncle and the master of Grantham School agreed that he should make better use of his obvious intelligence. They both persuaded Newton's mother to change her mind and she allowed him to prepare for a university education. By June 1661, Newton was committed to Trinity College at Cambridge University. He ignored the traditional course work, which was based on Greek philosopher Aristotle, and read many of the great scientific books of the day. Since he also found the work of Greek geometer Euclid of Alexandria to be"trifling""(trivial), he instead read the works of German astronomer and mathematician Johannes Kepler, French Algebraist and philosopher Rene Desciates, and Italian Galileo among many others. During his later years Newton enjoyed power and worldly success. Newton was elected president of the Royal Society in 1703 and was annually reelected until his death. In 1704 he published his second major work, the Opticks, based largely on work completed decades before. He was knighted in 1705. Although his creative years had passed, Newton continued to exercise a profound influence on the development of science. In effect, the Royal Society was Newton's instrument, and he played it to his personal advantage. . In the end, the
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Are you suffering in availing a loan due to being self employed? Have inadequate or not capable income as there is no fixed monthly income in your own business. In such critical times your homeowner position will aid you. Financial lenders have planned self employed secured loans that offer cash to self employed people only.
These loans are particularly prepared to offer financial support to the self employed people for satisfying their various fiscal needs on time. Before you apply for this credit facility, you should make sure that you have some property at your name. Due to the secured nature of the loan, you have to provide collateral against the loan amount.
By offering the collateral you are allowed to obtain the loan amount varying from £5,000 to £100,000 for the repayment tenure of 1 – 25 years. With the help of the amount you can meet your numerous demands on time. Even, you can use money for satisfying luxurious monetary needs as well. You can use the funds for the needs like clearing for old debts, car purchasing, home renovation, education purpose, etc.
Even if you are unemployed and can offer the security, you can apply for unemployed loans without any question. Credit rating plays the major role in deciding the lending process. Good credit score makes the procedure very simple because your good credit score shows your financial character.
However, bad credit ratings can also apply for this cash facility because lender gets against the loan. There is no matter that you have the default or bankruptcy in your report, but you have to pay slightly higher interest rate in comparison of normal borrowers.
Now if you are looking for the smooth approval, then you can apply through online medium. Online medium is easy and fast. You can also make some market search to avail the good scheme by comparing the various lenders. Lender provides the simple online application form that you require to complete for the procedure. Self employed secured loans offer the amount by taking the collateral. These loans are designed to give smooth financial help to self employed people without any hassle.


Find out more at Lexus.com.au/Corporate To access the Lexus Corporate Programme please contact your nearest Dealer (lexus.com.au/contact/find-a -dealer) or dedicated Lexus Corporate Sales Manager: 1. Conditions apply. See your Lexus dealer for further details. 2. Complimentar y servicing expires at 3 years or 60,000kms from the date of first registration, whichever occurs first. THE BENEFITS – Preferential Corporate Pricing1 – 3 year / 60,000kms complimentary servicing2 – Reduced delivery fee of $995 (ex GST) – Priority ordering and allocation – Encore Privileges Programme, including: - Ser vice loan cars or complimentary pickup/dropoff - Lexus DriveCare, providing 24-hour roadside assistance - Lexus Exclusive events Nataliya Yavtushenko E. nataliya.yavtushenko@lexus.com.au M. 0477 006 738 DESERVING OF YOUR SUCCESS EXCLUSIVE BENEFITS FOR QUEENSLAND LAW SOCIETY The Lexus Corporate Programme offers tailored perks for Queensland Law Society members. Whether your style is a luxury hatch, sedan, coupe or SUV, we can put you and your brand new Lexus in the luxury lane. Overseas model shown.


Even where petitioner disclosed additional income on assurance of survey party that his case would not be taken-up for scrutiny; Assessing Officer was still empowered to select petitioner's case for scrutiny assessment.
c) Subsequently, the petitioner was served with notice under section 143(2) informing him that his case was selected for scrutiny assessment. In the instant writ, the petitioner had challenged the legality and validity of scrutiny assessment.


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Q: Feature creation based on irregular time series with strictly monotonic values I am given the following problem to run predictive maintenance on the following mess:
I have some IoT sensors out in the field, but I cannot easily change the software on them. Also, data collection there is not cheap and thus, I am faced with the following problem of how to create useful features from the following messy data:
First, the sampling is highly irregular (sometimes minutes, sometimes only every couple of days, sometimes weeks between measurements).
Second, I mostly get cumulated data, i.e. strictly monotonic increases like:
This machine was in continuous operation for 0-1h: 15 times
1-2h: 267 times
2-5h: …
at the first instance of the sampling, in the next instance it might be:
0-1h: 24 times
1-2h: 345 times
...
I.e., Data accumulates and typical measures like the mean or frequencies are not directly applicable before differentiation, which did not work well for me.
How would you recommend pre-processing the data to take care of the irregularities and the strictly cumulated nature of the data, so that I could feed them into a simple machine learning algorithm (tree, linreg, ..)? What are the typical pitfalls? Literature recommendations?
An idea I have tried:
Normalize the increase between two sampling instances (i.e. (C(t_i)-C(t_i-1))/(t_i-t_i-1)) but it tends not to work well, as sometimes the difference is weeks and the differentiation is stable, but sometimes it is only hours and thus, very noisy. Adding additional smoothing drains out the signal more ...
Is there a canonical way to work with data like that? Is there a standard way to handle this?

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Grit Expectations: Councils and Road Authorities Get Ready For Winter
Mark Salisbury	 Jan 14, 2019
Driver Training/Safety, General News, News, Newsletter, Roads, Secondary News
With icy weather set to hit the UK later this week and into next week, and temperatures forecast to be well below zero, Transport Network – publishers of Highways and Smart Highways magazines and a leading highways and transport event company – has reviewed the nation's readiness for all that winter has to throw at it.
Dominic Browne, editor of Highway says: "Motorists will be pleased to hear the news that councils had 1.4 million tonnes of salt stockpiled in preparation for winter, with 94% of councils suggesting they have either more salt or the same level as last year. This means we should not see a return to the dark days of some eight years ago when highway authorities simply ran out of salt. Things have improved since then, with councils given specific resilience targets for their stockpiles. However issues still arise when individual areas suffer the brunt of the extreme weather.
"Last year, some councils ran out of salt stocks due simply to the logistics of getting fresh orders to depots on time. This was a combination of hold ups from suppliers, and a failure to prioritise those councils with the most urgent need.
"This year, the best advice on reorders, is the earlier they can the better, and to make sure they have back-up plans with alternative suppliers," he adds.
"According to the Local Government Association (LGA) annual Winter Readiness Survey, more than half (55%) of councils will share salt stocks, 30% will share gritting machinery, and one in five (22%) plan to team up across their boundaries to send staff to key locations. While this collaborative working is a step in the right direction, drivers who have been stuck in their cars in freezing conditions will want to know why these numbers aren't higher.
"Finally, last winter we saw some issues with crashes and pile-ups on steep inclines on roads, particularly in Scotland. Drivers will want to know that all roads authorities have identified those treacherous parts of their network, and have a plan and equipment such as heavy recovery vehicles in place should pile-ups occur."
Look out this winter for new gritters being used by Highways England on motorways and major roads. The bright orange vehicles use innovative technology, which includes being able to pre-programme them with information specific to each gritting route, spreading salt onto the road automatically, taking into account any specific requirements for bridges, landscape and other road features, and allowing the driver to focus on driving at all times.
However, technology in cars could play a stronger role in keeping roads moving. Paul Hutton, editor of Smart Highways says: "We're only just beginning to understand how connected vehicles can help us manage bad weather better. For example every car has a thermometer which, when connected, can be programmed to anonymously alert road authorities of locations where temperatures drop below a certain level. Similarly, when the vehicle's windscreen wipers are on high, or when a car skids, that information can be used to warn traffic control rooms and other drivers. Motoring organisations are connected to some of their members' vehicles to alert them to weather-related problems such as a struggling battery, in order to reduce the number of emergency breakdowns – but the number is still relatively small.
"Over time, these technological innovations will become commonplace, just as air conditioning, central locking or electrically operated windows are no longer restricted to premium cars. In the meantime, it pays off for motorists to carry warm spare clothes and boots in case of emergencies. And better still, to make sure their car is fully maintained to help head off emergencies in the first place."
Gritting Roads, Winter Driving


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A stable is a building in which livestock, especially horses, are kept.  It most commonly means a building that is divided into separate stalls for individual animals and livestock. There are many different types of stables in use today; the American-style barn, for instance, is a large barn with a door at each end and individual stalls inside or free-standing stables with top and bottom-opening doors.  The term "stable" is also used to describe a group of animals kept by one owner, regardless of housing or location.

The exterior design of a stable can vary widely, based on climate, building materials, historical period and cultural styles of architecture. A wide range of building materials can be used, including masonry (bricks or stone), wood and steel. Stables also range widely in size, from a small building housing one or two animals to facilities at agricultural shows or race tracks that can house hundreds of animals.

History
The stable is typically historically the second-oldest building type on the farm. The world's oldest horse stables were discovered in the ancient city of Pi-Ramesses in Qantir, in Ancient Egypt, and were established by Ramesses II (c. 1304–1213 BC). These stables covered approximately 182,986 square feet, had floors sloped for drainage, and could contain about 480 horses.
Free-standing stables began to be built from the 16th century. They were well built and placed near the house  because these animals were highly valued and carefully maintained. They were once vital to the economy and an indicator of their owners' position in the community. Relatively few examples survive of complete interiors (i.e. with stalls, mangers and feed racks) from the mid-19th century or earlier.

Traditionally, stables in Great Britain had a hayloft on their first (i.e. upper) floor and a pitching door at the front.  Doors and windows were symmetrically arranged.  Their interiors were divided into stalls and usually included a large stall for a foaling mare or sick horse. The floors were cobbled (or, later, bricked) and featured drainage channels. Outside steps to the first floor were common for farm hands to live in the building.

Horses

For horses, stables are often part of a larger complex which includes trainers, vets and farriers.

Other uses

The word stable is also used metonymically to refer to the collection of horses that the building contains (for example, the college's stable includes a wide variety of breeds) and even, by extension, metaphorically to refer to a group of people—often (but not exclusively) athletes—trained, coached, supervised or managed by the same person or organisation. For example, art galleries typically refer to the artists they represent as their stable of artists. Analogously, car enthusiast magazines sometimes speak of collectible cars in this way, referring to the cars in a collector's stable (most especially when the metaphor can play on the word association of pony cars).

Historically, the headquarters of a unit of cavalry, not simply their horses' accommodation, was known as a "stable".

Gallery

See also

Glossary of equestrian terms
Horse management
Livery stable
Nativity of Jesus
Pen

References

Agricultural buildings
Buildings and structures used to confine animals
Horse management

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\section{Introduction}

K. Saito \cite{Saito_TLDFLVF} introduced the notion of
the sheaf of logarithmic vector fields along a divisor
and proved that it is always reflexive.
A divisor $D$ in a variety $S$ is said to be {\em free}
if the sheaf of logarithmic vector field along $D$
is a free $\ensuremath{\mathcal{O}}_S$-module.
He proved that the discriminant in the parameter space
of the semi-universal deformation
of an isolated hypersurface singularity
is always free.

When the ambient space is the projective space $\ensuremath{\mathbb{P}}^\ell$,
an $\ensuremath{\mathcal{O}}_{\ensuremath{\mathbb{P}}^\ell}$-module is said to be free
if it is the direct sum $\bigoplus_{i} \ensuremath{\mathcal{O}}_{\ensuremath{\mathbb{P}}^\ell}(a_i)$
of invertible sheaves.
The problem of characterizing free divisors
in projective spaces has attracted much attention,
especially when the divisor is given
as an arrangement of hyperplanes.
See e.g. \cite{Terao_GE}.
If a divisor in $\ensuremath{\mathbb{P}}^\ell$ is free,
then the passage from the divisor
to the sheaf of logarithmic vector fields
causes loss of information;
only the sequence $\{ a_i \}_{i=1}^\ell$ of integers
is left,
and it is impossible to reconstruct the divisor
from this finite amount of information.

In the opposite extreme,
Dolgachev and Kapranov \cite{Dolgachev-Kapranov}
asked when the the sheaf $\ensuremath{\mathcal{T}}( - \log D)$
contains enough information
to reconstruct $D$.
A divisor $D$ in $\ensuremath{\mathbb{P}}^\ell$ is said to be {\em Torelli}
if the isomorphism class of $\ensuremath{\mathcal{T}}( - \log D)$
as an $\ensuremath{\mathcal{O}}_{\ensuremath{\mathbb{P}}^\ell}$-module
determines the divisor $D$.
Their main result is the condition
for an arrangement of sufficiently many hyperplanes
in $\ensuremath{\mathbb{P}}^\ell$ to be Torelli.

In this paper,
we discuss the case when $\ell = 2$
and $D$ is a smooth cubic curve.
Our main result asserts that
$D$ is Torelli precisely when the $j$-invariant of $D$ is not zero.
The strategy of our proof is the following:
\begin{enumerate}
\item
 The set of jumping lines of the sheaf of logarithmic vector fields
 along a smooth cubic curve coincides with its Cayleyan curve.
\item
 For a smooth cubic curve with a non-vanishing $j$-invariant,
 the Cayleyan curve determines the original curve
 up to three possibilities.
\item
 The set of ``jumping cubic curves'' fixes this left-over ambiguity
 and the Torelli property holds.
\item
 When the $j$-invariant of $D$ is zero,
 we can construct a family of divisors
 with isomorphic sheaves of logarithmic vector fields along them.
\end{enumerate}

Smooth cubic curves with vanishing $j$-invariants
provide examples of divisors
which are neither free nor Torelli.

{\bf Acknowledgment:}
We thank Igor Dolgachev for a stimulating lecture
in Kyoto in winter 2006
and Akira Ishii for valuable discussions and comments.
K. U. is supported
by Grant-in-Aid for Young Scientists (No.18840029).
M. Y. is supported by JSPS Postdoctoral Fellowship
for Research Abroad.

\section{Preliminaries}

\subsection{de Rham--Saito's lemma}

Let $A$ be a Noetherian ring and
$
 M = \bigoplus_{i=1}^n A e_i
$
be a free module over $A$ generated by
$e_1, \ldots, e_n$.
For $\omega_1, \ldots, \omega_r\in M$, put
$$
 \omega_1 \wedge \dots \wedge \omega_r
  = \sum_{1 \le i_1 < \dots <i_r \le n}
      a_{i_1, \dots, i_r} e_{i_1} \wedge \dots \wedge e_{i_r}.
$$
and define $\ensuremath{\mathfrak{a}}$ to be the ideal generated by
$a_{i_1, \cdots, i_r}$ for $1 \le r \le n$
and $1 \le i_1 < \dots <i_r \le n$.
We also define as follows:
\begin{eqnarray*}
 Z^p
  &=& \{ \varphi \in \wedge^p M \mid
         \omega_1 \wedge \dots \wedge \omega_r \wedge \varphi = 0 \}, \\
 B^p &=& \sum_{k=1}^r \omega_k \wedge (\wedge^{p-1} M), \\
 H^p &=& Z^p / B^p.
\end{eqnarray*}

\begin{theorem}[de Rham--Saito's lemma
\cite{de_Rham_SLDDF,Saito_GdRL}]
\begin{itemize}
\item[(1)] There exists an integer $\nu\in\ensuremath{\mathbb{Z}}_{\geq 0}$
such that $\ensuremath{\mathfrak{a}}^\nu H^p=0$ for
$0\leq p\leq n$.
\item[(2)] For $0\leq p< \operatorname{depth}_\ensuremath{\mathfrak{a}} A$,
we have $H^p=0$.
\end{itemize}
\end{theorem}

\subsection{Sheaf of logarithmic vector fields}

Let $A=\ensuremath{\mathbb{C}}[z_0, \ldots, z_\ell]$ be a polynomial ring
and $\mathrm{Der}_A$ be the module of $\ensuremath{\mathbb{C}}$-derivations
of $A$,
which is a free module of rank $\ell+1$;
$$
\mathrm{Der}_A=\sum_{i=0}^\ell A\frac{\partial}{\partial z_i}.
$$

\begin{definition}
\normalfont
For a homogeneous polynomial $f\in A$, we define
\begin{align*}
 D(-\log f)
  &= \{ \delta \in \mathrm{Der}_A \mid \delta f \in (f) \}, \\
 D_0(-\log f)
  &= \{ \delta \in \mathrm{Der}_A \mid \delta f = 0 \}.
\end{align*}
We put
$\operatorname{deg} z_i = 1$ and
$\operatorname{deg}(\partial / \partial z_i) = - 1$ for $i=0, \dots, l$.
The degree $k$ part of $D_0(-\log f)$
will be denoted by $D_0(-\log f)_k$.
\end{definition}

We have the direct sum decomposition
$$
 D(-\log f) = D_0(-\log f) \oplus A \cdot E,
$$
where
$$
 E = \sum_{i=0}^\ell z_i \partial/\partial z_i
$$
is the Euler vector field.
Let $\Omega_A$ be the module of differentials
$$
 \Omega_A^1=\bigoplus_{i=0}^\ell A dz_i,
$$
and $\Omega_A^k$ be its $k$-th exterior power
for $k = 0, \dots, \ell+1$.
We have an isomorphism of $A$-modules
$$
 D_0( - \log f)
  \cong \{ \omega \in \Omega^\ell \mid df \wedge \omega = 0 \}
$$
under the identification
$$
\begin{array}{rcl}
 \mathrm{Der}_A & \xrightarrow{\sim} & \Omega^\ell \\
 \rotatebox{90}{$\in$} & & \rotatebox{90}{$\in$} \\
 \sum_{i=0}^\ell f_i \frac{\partial}{\partial z_i} &
 \longmapsto &
 \sum_{i=0}^\ell (-1)^i f_i dz_0 \wedge \dots
   \wedge \widehat{dz_i} \wedge \dots \wedge dz_\ell.
\end{array}
$$
Let $D \subset \ensuremath{\mathbb{P}}^\ell$ be the hypersurface
defined by $f$.
If $D$ is smooth,
then the origin $0 \in \ensuremath{\mathbb{C}}^{l+1}$ is
the only zero locus of the Jacobi ideal
$$
 J(f) = \left(
         \frac{\partial f}{\partial z_0}, \dots,
         \frac{\partial f}{\partial z_\ell}
        \right),
$$
and hence we have
$$
 \operatorname{depth}_{J(f)} A = \ell + 1.
$$
Let $H^p$ be the $p$-th cohomology
of the complex
$$
 0
  \longrightarrow \Omega_A^0
  \xrightarrow{df\wedge} \Omega_A^1
  \xrightarrow{df\wedge} \cdots
  \xrightarrow{df\wedge} \Omega_A^{\ell}
  \xrightarrow{df\wedge} \Omega_A^{\ell+1}
  \longrightarrow 0.
$$
If $D$ is smooth,
then we have $H^p=0$ for $p = 0, \dots, \ell$
by de Rham--Saito's lemma.
Since
$$
 D_0(- \log f)
  \cong \operatorname{Ker} \left(
              df \wedge : \Omega^{\ell} \to \Omega^{\ell+1}
             \right),
$$
the sequence
\begin{equation} \label{eq:res_Dlog}
 0
  \longrightarrow \Omega_A^0
  \xrightarrow{df\wedge} \Omega_A^1
  \xrightarrow{df\wedge} \cdots
  \xrightarrow{df\wedge} \Omega_A^{\ell-1}
  \xrightarrow{df\wedge} D_0(- \log f)
  \longrightarrow 0
\end{equation}
gives a free resolution of $D_0(- \log f)$.
\medskip

The Euler sequence
$$
\begin{array}{ccccccccc}
 0 & \longrightarrow & \ensuremath{\mathcal{O}} & \longrightarrow & \ensuremath{\mathcal{O}}(1)^{\ell+1}
   & \longrightarrow & \ensuremath{\mathcal{T}}_{\ensuremath{\mathbb{P}}^{\ell}} & \longrightarrow & 0, \\
 & & \rotatebox{90}{$\in$} & & \rotatebox{90}{$\in$} & & & & \\
 & & 1 & \mapsto & E & & & &
\end{array}
$$
shows that the sheafification $\scT_{\bP^\ell}(-\log f)$ of ${D_0(-\log f)}$
can be considered
as a subsheaf of the tangent sheaf $\ensuremath{\mathcal{T}}_{\ensuremath{\mathbb{P}}^\ell}$;
$$
 \scT_{\bP^\ell}(-\log f) \subset \ensuremath{\mathcal{T}}_{\ensuremath{\mathbb{P}}^{\ell}}.
$$
It is the sheaf of holomorphic vector fields
tangent to the hypersurface $D$ at smooth points of $D$.
If $D$ is smooth,
we have the short exact sequence
$$
 0 \longrightarrow \scT_{\bP^\ell}(-\log f) \longrightarrow \ensuremath{\mathcal{T}}_{\ensuremath{\mathbb{P}}^\ell}
   \longrightarrow \ensuremath{\mathcal{N}}_{D/\ensuremath{\mathbb{P}}^\ell} \longrightarrow 0,
$$
where $\ensuremath{\mathcal{N}}_{D/\ensuremath{\mathbb{P}}^\ell}$ is the normal bundle.
We have an isomorphism
$$
 d f|_D : \ensuremath{\mathcal{N}}_{D / \ensuremath{\mathbb{P}}^{\ell}} \xrightarrow{\sim} \ensuremath{\mathcal{O}}_D(d),
$$
where
$$
 d = \operatorname{deg} f.
$$

If $D$ is smooth,
then the sheaf $\ensuremath{\mathcal{T}}_{\ensuremath{\mathbb{P}}^\ell}(-\log f)$ has the resolution
\begin{equation} \label{eq:res_Tlog}
 0 \to \ensuremath{\mathcal{O}}(1-(d-1)\ell) \to \dots
   \to \ensuremath{\mathcal{O}}(3-2d)^{\oplus \binom{\ell+1}{\ell-2}}
   \to \ensuremath{\mathcal{O}}(2-d)^{\oplus \binom{\ell+1}{\ell-1}}
   \to \scT_{\bP^\ell}(-\log f)
   \to 0
\end{equation}
obtained by sheafifying the exact sequence
(\ref{eq:res_Dlog}).
We also have
$$
 \Gamma
  \left(
   \ensuremath{\mathbb{P}}^\ell, \scT_{\bP^\ell}(-\log f)(k)
  \right)
 = D_0(-\log f)_k
$$
for $k \in \ensuremath{\mathbb{Z}}$.

\section{Plane curves}

Now we set $\ell=2$
to focus our attention on plane curves.
Let
$
 f \in \ensuremath{\mathbb{C}}[z_0, z_1, z_2]
$
be a homogeneous polynomial of degree $d$ and
$D \subset \ensuremath{\mathbb{P}}^2$ be the curve
defined by $f$.
Define $\ensuremath{\mathcal{F}}$ as the cokernel of
$
 df \wedge : \ensuremath{\mathcal{O}}(3 - 2 d) \to \ensuremath{\mathcal{O}}(2 - d)^{\oplus 3}
$
so that we have the exact sequence
\begin{equation} \label{eq:exact}
 0
  \longrightarrow \ensuremath{\mathcal{O}}(3 - 2 d)
  \xrightarrow{df\wedge} \ensuremath{\mathcal{O}}(2 - d)^{\oplus 3}
  \longrightarrow \ensuremath{\mathcal{F}}
  \longrightarrow 0.
\end{equation}
The Chern polynomial of $\ensuremath{\mathcal{F}}(k)$ is given by
\begin{align*}
 c_t(\ensuremath{\mathcal{F}}(k))
  &:= 1 + c_1(\ensuremath{\mathcal{F}}(k)) t + c_2(\ensuremath{\mathcal{F}}(k)) t^2 \\
  &= c_t(\ensuremath{\mathcal{O}}(2 - d + k))^3 c_t(\ensuremath{\mathcal{O}}(3 - 2 d + k))^{-1} \\
  &= 1 + (3 - d + 2 k) t + (d^2 - 3 d + 3 + k^2 + (3 - d) k)t^2
\end{align*}
for $k \in \ensuremath{\mathbb{Z}}$.
If $D$ is smooth,
then we have
\begin{align*}
 \ensuremath{\mathcal{F}}
  &:= \operatorname{Coker}
   (df \wedge : \ensuremath{\mathcal{O}}(3 - 2 d) \to \ensuremath{\mathcal{O}}(2 - d)^{\oplus 3}) \\
  &\cong \operatorname{Coim}
   (df \wedge : \ensuremath{\mathcal{O}}(2 - d)^{\oplus 3} \to \ensuremath{\mathcal{O}}(1)^{\oplus 3}) \\
  &\cong \operatorname{Im}
   (df \wedge : \ensuremath{\mathcal{O}}(2 - d)^{\oplus 3} \to \ensuremath{\mathcal{O}}(1)^{\oplus 3}) \\
  &\cong \operatorname{Ker}
   (df \wedge : \ensuremath{\mathcal{O}}(1)^{\oplus 3} \to \ensuremath{\mathcal{O}}(d)) \\
  &\cong \scT_{\bP^\ell}(-\log f).
\end{align*}

\begin{lemma} \label{lem:stability}
If $D$ is smooth,
then $\ensuremath{\mathcal{T}}_{\ensuremath{\mathbb{P}}^2}(-\log f)$ is stable.
\end{lemma}

\begin{proof}
We consider $\ensuremath{\mathcal{F}}([(d - 3) / 2])$ instead of $\ensuremath{\mathcal{T}}_{\ensuremath{\mathbb{P}}^2}(- \log f)$
whose first Chern number is normalized to either
$0$ (when $d$ is odd) or $-1$ (when $d$ is even).
Then $\ensuremath{\mathcal{F}}([(d - 3) / 2])$ is stable
if and only if it has no global section.
This follows from the cohomology long exact sequence
associated with the short exact sequence (\ref{eq:exact})
tensored with $\ensuremath{\mathcal{O}}_{\ensuremath{\mathbb{P}}^2}([(d - 3) / 2])$.
\end{proof}

\section{Smooth cubic curves}

Let $f \in \ensuremath{\mathbb{C}}[z_0, z_1, z_2]$ be a homogeneous polynomial
of degree three and $D \subset \ensuremath{\mathbb{P}}(V)$ be a cubic curve
defined by $f$,
where $V = \operatorname{Spec} \ensuremath{\mathbb{C}}[z_0, z_1, z_2]$.
We assume that $D$ is smooth.

\subsection{Jumping lines}

Let $L$ be a point in the dual projective plane $\ensuremath{\mathbb{P}}(V^*)$
defined by a linear form
$
 \alpha = \alpha_0 z_0 + \alpha_1 z_1 + \alpha_2 z_2 \in V^*.
$
We can think of $L$ as a line in $\ensuremath{\mathbb{P}}(V)$.
Restricting the short exact sequence (\ref{eq:exact}) to $L$
and taking the cohomology long exact sequence,
we have
$$
0
 \longrightarrow H^0(\ensuremath{\mathcal{F}}|_L)
 \longrightarrow H^1(\ensuremath{\mathcal{O}}_L(-3))
 \longrightarrow H^1(\ensuremath{\mathcal{O}}_L(-1))^3
 \longrightarrow H^1(\ensuremath{\mathcal{F}}|_L)
 \longrightarrow 0.
$$
Since
$$
 H^1(\ensuremath{\mathcal{O}}_L(-3)) \cong H^0(\ensuremath{\mathcal{O}}_L(1))^* \cong \ensuremath{\mathbb{C}}^2
$$
and
$$
 H^1(\ensuremath{\mathcal{O}}_L(-1)) \cong H^0(\ensuremath{\mathcal{O}}_L(-1))^* = 0,
$$
we have
$$
 \dim H^0(\ensuremath{\mathcal{F}}|_L) =2.
$$
Hence $\ensuremath{\mathcal{F}}|_L$ is either
$$
 \ensuremath{\mathcal{F}}|_L =
  \begin{cases}
   \ensuremath{\mathcal{O}}_L \oplus \ensuremath{\mathcal{O}}_L & \text{$L$ is generic}, \\
   \ensuremath{\mathcal{O}}_L(-1) \oplus \ensuremath{\mathcal{O}}_L(1) & \text{$L$ is a jumping line.}
  \end{cases}
$$
In particular,
$$
 \text{$L$ is a jumping line}
  \Longleftrightarrow H^0(\ensuremath{\mathcal{F}}(-1)|_L)\neq 0.
$$
By tensoring $\ensuremath{\mathcal{O}}_L(-1)$ with
the short exact sequence (\ref{eq:exact})
and taking the cohomology long exact sequence,
we have
$$
\begin{array}{ccccccccccc}
0  &
\rightarrow  &
H^0(\ensuremath{\mathcal{F}}(-1)|_L)  &
\rightarrow  &
H^1(\ensuremath{\mathcal{O}}_L(-4))&
\xrightarrow{df\wedge} &
H^1(\ensuremath{\mathcal{O}}_L(-2)^{\oplus 3})&
\rightarrow  &
H^0(\ensuremath{\mathcal{F}}(-1)|_L)&
\rightarrow  &
0\\
&&&&\rotatebox{90}{$\cong$}&&\rotatebox{90}{$\cong$}&&&&\\
&&&& H^0(\ensuremath{\mathcal{O}}_L(2))^*

&H^0(\ensuremath{\mathcal{O}}_L^{\oplus 3})^*.&&&&
\end{array}
$$
Since
$
 H^0(\ensuremath{\mathcal{O}}(2)|_L)
  \cong \operatorname{Sym}^2V^*/(z_0 \alpha, z_1 \alpha, z_2 \alpha),
$
the set $S = S(\scT_{\bP^\ell}(-\log f)) \subset \ensuremath{\mathbb{P}}(V^*)$
of jumping lines
is characterized as follows;
\begin{align}
 L \in S
 & \Longleftrightarrow
  \text{$(df \wedge)^* : H^0(\ensuremath{\mathcal{O}}_L^{\oplus 3}
   \to H^0(\ensuremath{\mathcal{O}}_L(2)))$ is not an isomorphism}
     \nonumber \\
 & \Longleftrightarrow
  \text{
   $z_0 \alpha, z_1 \alpha, z_2 \alpha,
    \partial_0 f, \partial_1 f, \partial_2 f$
   are linearly dependent in $\operatorname{Sym}^2 V^*$}.
     \label{eq:jumping_line}
\end{align}

\subsection{Cayleyan curves}

Here we prove the following:

\begin{proposition} \label{prop:jumping_line-Cayleyan}
Let $D \subset \ensuremath{\mathbb{P}}(V)$ be a smooth cubic curve
defined by a polynomial $f$.
Then the set $S = S(\scT_{\bP^\ell}(-\log f)) \subset \ensuremath{\mathbb{P}}(V^*)$
of jumping lines of $\scT_{\bP^\ell}(-\log f)$
in the dual projective plane $\ensuremath{\mathbb{P}}(V^*)$
is the Cayleyan curve of $D$.
\end{proposition}

First we recall the definition of the Cayleyan curve
of a plane cubic curve
following Artebani and Dolgachev \cite{Artebani-Dolgachev}.
The {\em first polar} of a plane curve $D = \{ f = 0 \}$
with respect to a point $q = [a_0 : a_1 : a_2] \in \ensuremath{\mathbb{P}}(V)$
is the curve
$
 P_q(D)
  = \{
     a_0 \partial_0 f + a_1 \partial_1 f + a_2 \partial_2 f = 0
    \}$
whose degree is one less than that of $D$.
One can show that when $D$ is a cubic curve,
the Hessian curve
$
 \operatorname{He}(D)
  = \{
     \det \left( (\partial_i \partial_j f )_{i,j=1}^3 \right) = 0
    \} \subset \ensuremath{\mathbb{P}}(V)
$
consists of points $q \in \ensuremath{\mathbb{P}}(V)$
such that the polar curve $P_q(D)$ decomposes
into the union of two lines.
For $q \in \operatorname{He}(D)$,
let $s_q \in \ensuremath{\mathbb{P}}(V)$ be the singular point of $P_q(D)$
and $L_q \in \ensuremath{\mathbb{P}}(V^*)$ be the line
connecting $q$ and $s_q$.
It is known that $s_q$ lies on $\operatorname{He}(D)$
and the map
$$
\begin{array}{ccccc}
 s & : & \operatorname{He}(D) & \longrightarrow & \operatorname{He}(D) \\
 & & \rotatebox{90}{$\in$} & &  \rotatebox{90}{$\in$} \\
 & & q & \mapsto & s_q
\end{array}
$$
is a fixed-point-free involution on $\operatorname{He}(D)$.
The image of the map
$$
\begin{array}{ccc}
 \operatorname{He}(D) & \longrightarrow & \ensuremath{\mathbb{P}}(V^*) \\
 \rotatebox{90}{$\in$} & &  \rotatebox{90}{$\in$} \\
 q & \mapsto & L_q
\end{array}
$$
is called the {\em Cayleyan curve} of $D$,
which is known to be the quotient of $\operatorname{He}(D)$
by the involution $s$.
A linear form
$
 \alpha = \alpha_0 z_0 + \alpha_1 z_1 + \alpha_2 z_2 \in V^*
$
represents a point in the Cayleyan curve of $D$
if and only if there is a point $[a_0 : a_1 : a_2] \in \ensuremath{\mathbb{P}}^2$
such that
$$
 a_0 \partial_0 f + a_1 \partial_1 f + a_1 \partial_1 f
  \in \alpha \cdot V^*.
$$
This is precisely the condition (\ref{eq:jumping_line})
for the line $[\alpha] \in \ensuremath{\mathbb{P}}(V^*)$
to be a jumping line of $\scT_{\bP^\ell}(-\log f)$.

\subsection{The set of jumping lines and $j$-invariant}

Here we prove the following:

\begin{proposition}
Let $D$ be the smooth cubic curve
defined by a polynomial $f$.
Then the set $S(\scT_{\bP^\ell}(-\log f))$ of jumping lines is singular
if and only if the $j$-invariant of $D$ is zero.
\end{proposition}

\begin{proof}
Choose a coordinate of $V$
so that $f$ is a Hesse cubic
\begin{equation} \label{eq:Hesse}
 f_t(z_0, z_1, z_2)
  = z_0^3 + z_1^3 + z_2^3 - 3 t z_0 z_1 z_2,
\end{equation}
where $t \in \ensuremath{\mathbb{C}} \setminus \{ 1, \zeta, \zeta^2 \}$
and $\zeta = \exp[2 \pi \sqrt{-1} / 3]$.
Recall that $D = \{ f_t = 0 \} \subset \ensuremath{\mathbb{P}}^2$ is smooth
if and only if $t^3 \ne 1$.
The set $S = S(\scT_{\bP^\ell}(-\log f))$ of jumping lines,
which coincides with the Cayleyan curve of $D$,
is a Hesse cubic
$$
 t (\alpha_0^3 + \alpha_1^3 + \alpha_2^3)
  - (t^3 + 2)  \alpha_0 \alpha_1 \alpha_2 = 0
$$
in the dual projective plane.
It is the union of three lines in general position
if $t=0$ or $(3 t)^3 = (t^3 + 2)^3$.
Since
$$
 (t^3 + 2)^3 - (3 t)^3 = (t^3 - 1)^2 (t^3 + 8)
$$
and the $j$-invariant $j(D)$ of $D$ is given by
$$
 j(D) = \frac{1}{64} t^3 \frac{(t^3 + 8)^3}{(t^3 - 1)^3},
$$
the Cayleyan curve of $D$ is smooth if and only if $j(D) \ne 0$,
and decomposes into the union of three lines in general position
if $j(D) = 0$.
\end{proof}

\subsection{Restricting $\scT_{\bP^\ell}(-\log f)$ to other cubic curves}

Here we consider the restriction of the sheaf $\scT_{\bP^\ell}(-\log f)$
to another cubic curve $E$
defined by a polynomial $g$.
From the exact sequence (\ref{eq:exact}),
we have
$$
 0 \longrightarrow \ensuremath{\mathcal{O}}(-3)|_E
   \longrightarrow \ensuremath{\mathcal{O}}(-1)^{\oplus 3}|_E
   \longrightarrow \ensuremath{\mathcal{F}}|_E
   \longrightarrow 0.
$$
Hence we have
$$
\begin{array}{ccccccccccc}
0  &
\rightarrow  &
H^0(\ensuremath{\mathcal{F}}|_E)  &
\rightarrow  &
H^1(\ensuremath{\mathcal{O}}(-3)|_E)&
\stackrel{df\wedge}{\longrightarrow}  &
H^1(\ensuremath{\mathcal{O}}(-1)^{\oplus 3}|_E)&
\rightarrow  &
H^0(\ensuremath{\mathcal{F}}|_E)&
\rightarrow  &
0\\
&&&&\rotatebox{90}{$\cong$}&&\rotatebox{90}{$\cong$}&&&&\\
&&&& H^0(\ensuremath{\mathcal{O}}(3)|_E)^*
&
&H^0(\ensuremath{\mathcal{O}}(1)^{\oplus 3}|_E)^*.&&&&
\end{array}
$$
Since $H^0(\ensuremath{\mathcal{O}}(3)|_E)=\operatorname{Sym}^3V^*/(g)$ and
$H^0(\ensuremath{\mathcal{O}}(1)|_E)^3=(V^*)^3$,
the map $df \wedge$ is dual to the map induced by
$$
\begin{array}{ccc}
 (V^*)^3 & \longrightarrow & \operatorname{Sym}^3 V^* \\
 \rotatebox{90}{$\in$} & & \rotatebox{90}{$\in$} \\
 (F_0, F_1, F_2) & \longrightarrow &
  F_0 \partial_0 f + F_1 \partial_1 f + F_2 \partial_2 f.
\end{array}
$$
This map is injective due to de Rham--Saito's lemma,
and the image can be identified with the degree $3$ part
$J(f)_3$ of the Jacobi ideal.
Hence we have
$$
 H^0(\ensuremath{\mathcal{F}}|_E) =
  \begin{cases}
   \ensuremath{\mathbb{C}} & g \in J(f)_3, \\
   0 & g \not \in J(f)_3.
  \end{cases}
$$
By an explicit calculation,
we obtain the following:
\begin{proposition} \label{prop:jumping_cubic}
Let $f_t$ be the Hesse cubic in (\ref{eq:Hesse})
and put
$$
 g = \sum_{0 \le i \le j \le k \le 2} a_{ijk} z_i z_j z_k.
$$
Then the hyperplane $J(f_t)_3 \subset \operatorname{Sym}^3 V^*$
is given by
$$
 J(f_t)_3 = \{ g \mid a_{012} + t (a_{000} + a_{111} + a_{222}) = 0 \}.
$$
\end{proposition}

\section{Torelli theorem}

Here we prove our main result:

\begin{theorem}
Let $C$ and $C'$ be smooth cubic curves
with non-vanishing $j$-invariants.
If $\ensuremath{\mathcal{T}}(-\log C)$ is isomorphic to $\ensuremath{\mathcal{T}}(-\log C')$
as an $\ensuremath{\mathcal{O}}_{\ensuremath{\mathbb{P}}^2}$-module, then $C = C'$.
\end{theorem}

\begin{proof}
Take a homogeneous coordinate
of the dual projective plane
so that the set of jumping lines of $\ensuremath{\mathcal{T}}(- \log C)$
is a Hesse cubic.
Since a smooth cubic
whose Cayleyan curve is a smooth Hesse cubic
must be a Hesse cubic,
$C$ and $C'$ are Hesse cubics.
Then Proposition \ref{prop:jumping_cubic} shows that
$C$ must coincide with $C'$.
\end{proof}

\begin{remark}
The Torelli theorem fails for cubic curves
with vanishing $j$-invariants.
Indeed,
the family
$$
 a z_0^3 + b z_1^3 + c z_0^3 = 0, \qquad a, b, c \in \bC^{\times}
$$
consists of cubic curves
with identical Cayleyan curves given by
$$
 \alpha_0 \alpha_1 \alpha_2 = 0.
$$
Since the set of jumping lines determines
a unique stable bundle
if it consists of three lines in general position
by Barth \cite{Barth_MVBPP},
the sheaf of logarithmic vector fields
does not depend on $a$, $b$, and $c$.
\end{remark}

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You are here: Home Newsroom News 50 – the age for vitality Boots UK reveals the 'F Factor' attitudes of this get-up-and-go generation
50 – the age for vitality Boots UK reveals the 'F Factor' attitudes of this get-up-and-go generation
News Brand
Feeling younger than ever before, nearly half a million women in the UK, including Davina McCall, Kate Garraway and Ulrika Jonsson, will turn fifty in 2017.1 New research reveals that far from feeling past their heyday, 4 in 5 (82%) are winding up, not down as they reach this age landmark, however nine out of ten women (92%) approaching the big 5-0 have concerns about their health. Boots UK is raising the health and happiness of the nation by offering health and lifestyle advice to give these women the knowledge, tools and confidence to help them reach their expanding life ambitions. By identifying the three 'F Factor' attitudes women have towards turning fifty, Health Psychologist Dr Linda Papadopoulos and Boots Pharmacist Liz McPherson offer hints and tips to feeling fabulous at fifty.
Health Psychologist Dr Linda Papadopoulos says, "Women turning fifty now feel younger than their previous generations, with most viewing this time in their lives as a threshold of new opportunities and experiences. That said, turning fifty heralds a decade of transition, with many physical and psychological changes on the horizon, and there are some women who are feeling less certain about their next life chapter. That's why I'm working with Boots UK to identify three main typologies of women approaching fifty, so they can offer advice appropriate to how women really feel and behave towards their health at this milestone age."
THE THREE F FACTOR TYPOLOGIES
FEARLESS (40% of women): Nothing can stop her
"A fearless type lives life to the max. Facing the future with excitement and positivity – she's first in line to take-up a new dance class, start a new career, or travel the globe. Caring about her appearance as much as her health, she feels like nothing can stop her when it comes to trying new things."
FATALISTIC (39% of women): What will be will be
"A fatalistic woman lives by the motto 'what will be will be'. She takes the ageing process in her stride, but can be complacent when it comes to looking after herself, ignoring health niggles rather than proactively looking to improve them."
FRETFUL (17% of women): Natural worrier
"A fretful type is a natural worrier. She feels anxious about what getting older will bring and finds it difficult to embrace change. She is very aware of how her body and health are changing, but might bottle up her worries rather than dealing with them proactively."
F FACTORS HEALTH ADVICE
Using these different attitudes of women approaching fifty, 55-year-old Boots Pharmacist Liz McPherson has created tailored health advice, drawing on her personal and professional experience. She encourages those facing into fifty to 'ASK' (Always, Start, Know) when encountering health concerns at this age:
Nothing can stop her
FATALISTIC
What will be will be
FRETFUL
Natural worrier
…know your limits and don't push your body too hard. Over exertion could see you miss out on the things you enjoy so be sensible about what you can manage."
…monitor your body and emotions for any notable changes and seek support rather than ignoring them."
…ask for help and advice if a health niggle is concerning you."
…having your free NHS Health Checks every five years, available to those aged 40-74. These will identify whether you're at higher risk of getting certain health problems, such as heart disease, diabetes, kidney disease and stroke."
…being proactive in future-proofing your health – small changes to your lifestyle can make a big difference to how your body manages age-related changes."
…doing, not worrying. If you are concerned about a health issue take action to improve it - worrying won't help."
…your local pharmacist is a great source of advice – drop in and ask about the best ways to support your active lifestyle."
…you don't have to simply accept the changes your body is experiencing. For many niggles associated with ageing there is support and treatment available."
…your real risks by taking advantage of the free online health assessments offered at Boots.com."
Boots Pharmacist Liz McPhersonalsogives her top tips on dealing with some of the top health worries of this generation:
Joint health: "Leading a healthy lifestyle is an easy way to help maintain joint health. Regular exercise will help strengthen muscles that support joints and keep joints flexible. You might also want to consider a supplement tailored for joint health.
Healthy diet: 50 is also a time when, more than ever, you can benefit from a healthier diet and lifestyle to ensure you enjoy life to the full."
Eye sight: "Changing eyesight at the age of fifty is very common it's important to have regular eye tests and take action when you notice any changes to your vision, so visit Boots Opticians for an eye test and advice on frames that suit you."
Energy levels and immunity: "Consider taking a multi-vitamin supplement containing vitamins B and C to help with energy levels or a supplement containing Zinc to help maintain a healthy immune system."
Despite the expected health niggles, women of this generation are still living their lives to the full, and are embracing what the coming years will bring. As they approach fifty, over a quarter (27%) plan to travel and see more of the world, and 1 in 5 (19%) are doing more exercise than ever before. A fifth of those surveyed have exciting career aspirations they want to achieve (19%) and 1 in 7 (15%) are looking to do more charity work or volunteering. Becoming more tech-savvy is a goal for 18% and 7% even want to get a tattoo.
Approach fifty feeling better than ever with advice and support from Boots UK.
For more information or additional quotes from Boots Pharmacist Liz McPherson, please contact the Boots UK Press Office:
Boots UK Press Office: 020 7025 6657/ bootshealthteam@redconsultancy.com
Research commissioned by Boots, surveying 1,002 women turning 50 in 2017 - November 2016, data on file
Additional quotes from Dr Linda Papadopoulos:
Typology One - Fearless (40%): "This woman embraces change rather than fearing it. Her identity is something that she is clear about and comfortable with and so she is adaptable when faced with change in either her circumstances, her appearance or even her health. She doesn't fear the future; she feels excitement at the prospect of new challenges rather than worry or anxiety. She is generally a glass half full type of person, approaching challenges with confidence and positivity - she is open to new ideas, thirsty to learn and always ready for new experiences – she's prime to be first in line to take-up a new dance class, do that degree she always aspired to do, or learn a new language for an exotic trip away."
Typology Two - Fatalistic (39%): "This woman deals with change by accepting that it's inevitable. She accepts that she cannot control all aspects of her life at all times and as a consequence takes setbacks, changes and surprises in her stride. She sees changes in her health and appearance as a normal part of life and as such can be complacent about her self-care. However, the fact that she is fatalistic when it comes to growing older means that she doesn't fear growing older but rather has a positive view of ageing, she doesn't feel that she has to rile against it but rather live the best life she can within the limits of the changes that growing older and entering a new life stage brings."
Typology Three - Fretful (17%): "This woman is a natural worrier. She feels that anticipating and worrying about things will make her more prepared to deal with them and as such is likely to be hyper-aware of changes in her health and overall functioning generally. While having some degree of awareness of her changing body is good, as it will instigate positive self-care processes, the downside is that this anticipatory anxiety will get in the way of her embracing change in a positive and productive way."
Boots UK (boots.com), the UK's leading pharmacy-led health and beauty retailer, is part of the Retail Pharmacy International Division of Walgreens Boots Alliance, Inc. (Nasdaq: WBA), the first global pharmacy-led, health and wellbeing enterprise. With 2,510**** stores in the UK, Boots UK is committed to providing exceptional customer and patient care, be the first choice for pharmacy and healthcare and offer innovative 'only at Boots' exclusive products such as the UK's leading skincare brand No7, all delivered with the great value customers love. Created over 165 years ago, the Boots brand is still at the heart of the communities it serves.
1. ONS Population Data, MYE2: Population estimates by single year of age and sex for local authorities in the UK, mid-2015, https://www.ons.gov.uk/peoplepopulationandcommunity/populationandmigration/populationestimates/datasets/populationestimatesforukenglandandwalesscotlandandnorthernireland, Accessed 06.12.16
2. NHS Choices, Menopause, http://www.nhs.uk/Conditions/Menopause/Pages/Introduction.aspx, Accessed 06.12.16
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Home 2020 April 14 World Ecuador: hundreds of corpses of Covid-19 victims left on streets and homes
April 14, 2020 | by	Math | 0
Nearly 800 bodies of coronavirus victims were removed from Guayaquil accommodation by a special force created by the government. The city is overwhelmed by its dead.
Guayaquil, the economic capital of Ecuador, is suffering from devastating chaos in the management of the new coronavirus, and has become one of the Latin American cities most affected by the pandemic.
For days, hundreds of corpses were left at home, even in the streets, wrapped in black plastic. Funeral services were overwhelmed and the health sector, lacking funds and personnel, collapsed.
The stench of death floats around hospitals. In front of the cemeteries are formed long lines of vehicles loaded with cardboard coffins.
An official announced on Sunday (April 12th) that nearly 800 corpses had been removed from Guayaquil housing by a special force made up of police and soldiers and created by the government.
"The number (of bodies) we have collected in homes with the special force has exceeded 700," said force chief Jorge Wated. Shortly after, this official indicated on his Twitter account that the number of corpses collected in homes was now 771, to which are added 631 bodies which were in hospitals, whose morgues are full.
And the worst is yet to come: the authorities of this small country of 17.5 million inhabitants predict up to 3,500 deaths from the Covid-19.
Lots of exchanges with Spain and Italy
The province of Guayas, of which Guayaquil is the capital, concentrates 73% of the nearly 7,500 cases, including at least 333 deaths confirmed Sunday at the national level since February 29.
"The mistakes made cannot be repeated! " Said Vice President Otto Sonnenholzner, who hails from this city of 2.7 million people, and who leads the government's fight against the virus.
The port of Guayaquil, on the Pacific coast (south-west), appeared from the start as a particularly vulnerable point: it was there that the first Ecuadorian case was detected, an old lady returned from Spain.
In Spain and Italy, countries very affected by the pandemic, live half a million Ecuadorians, many of whom migrated during the financial crisis of the 1990s.
The flow of exchanges between Guayaquil, Europe and also the United States is intense, especially in February and March, during school holidays.
Ecuador "reacted late," Daniel Simancas, epidemiologist at the Equinoxial Technological University (UTE) in Quito, told AFP.
Lack of strategy and delays
This had the "devastating consequences we have seen. The authorities themselves have apologized for the lack of strategies in the management of corpses, of forecasting in the supply of medical materials, " he added.
There were also delays in purchasing tests, with a weak epidemiological surveillance plan. The "broth" social Guayaquil aggravated the crisis.
Although Guayas is the most productive province in the country, its capital has a poverty rate of 11.2%, according to official figures for December.
Unemployment and underemployment affect 20% of the working population. "People want to go out to work and this is due to (…) informal employment," economist Alberto Acosta Burneo of the Spurrier firm told AFP.
Sociologist Carlos Tutiven, of Casa Grande University, also from Guayaquil, points to the "model of economic development" of this city, which resisted the socialism of ex-president Rafael Correa (2007-2017), made of social investments and high debt.
No formula has "been powerful enough to solve the inequity" where rich villas and shanty towns are located.
In Guayas, nearly 3,300 people have violated the government-imposed 15-hour curfew.
Despite the soldiers deployed in the streets, it is common to see street vendors there, without masks, and queues in front of the shops, without the recommended distance.
Many inhabitants with very precarious conditions
The sociologist highlights a "weakness" of information on precautionary measures, adding that "not everyone can be obedient or disciplined because the vast majority live in very precarious conditions" .
"Locking yourself in a four square meter apartment, a room where there are four, five, six people, is suffocating," he told AFP.
According to the UTE epidemiologist, the poor were not the only ones who did not stay at home: "Many families with a lot of money and power have underestimated the power of this virus and have not not respected quarantine measures " .
Much criticized, the government of President Lenin Moreno and the local authorities have admitted serious flaws.
Mayor Cynthia Viteri, herself contaminated and recovered, acknowledged the guilt of "everyone" .
"We see our dead fall in silence every day; we hear the neighbor cry for her deceased loved one (…) a pregnant woman who has no hospital to give birth to, and a hundred people have died from being unable to do dialysis. "
"Here, it is not only the health system that has collapsed, but also the funeral services and the morgues," she said.
Filed in: World
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Permanent wires behind teeth make flossing difficult - what can I do?
Hi, I have a question, I've recently had braces and now have permanent wires behind my teeth, this makes flossing difficult, is there a solution for this as I feel like I get a build up of plaque that I can no longer access? Thanks, Adam.
Dear Adam, Retention is very important after orthodontic treatment. You have removable and fixed options for retainers and sometimes both. The normal fixed straight wire can stop floss from going down between teeth and make it harder to keep a good oral hygiene. You should still be able to fit something like a small interdental brush through if not there might be glue stuck from when the retainer was fitted.
Have you had oral hygiene instructions in how to clean around the retainer? If not make an appointment with your dentist or hygienist to arrange for this to happen.
An electric toothbrush is always the best option to keep the fixed retainers clean.


Hanazuki loves being a Moonflower! With a little help from the Hemka, Sleepy Unicorn, and Dazzlessence Jones, she tries her best to keep her moon in balance and keep the Big Bad away. When two comets collide above them, Hanazuki's side of the moon is plunged into darkness. What's a Moonflower to do besides light a spark in the dark?


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Three New Deputy Rectors
Dr Magdalena Kalińska, Dr Bartłomiej Chludziński, Dr. habil. Aleksandra Derra, NCU Prof.
The NCU Rector, Prof. Andrzej Sokala, has appointed three Deputy Rectors: for equal treatment; for the safety of students and doctoral students; and an academic ombudsman. The Deputy Rectors will carry out their duties in the term of office 2020 - 2024.
Dr. habil. Aleksandra Derra, NCU Prof. from the Faculty of Philosophy and Social Sciences, was appointed Deputy Rector for equal treatment. Her task is to implement the principle of equal treatment at the University, develop a culture of respect for diversity and care for inclusion in all areas of the activities (scientific research, education, intra-institutional relations, relations with the social and business circles), so that the Nicolaus Copernicus Copernicus University fully realises its scientific potential and the obligations enshrined in the Nicolaus Copernicus University's Mission and Strategy and resulting from participation in the IDUB and YUFE projects.
Dr Magdalena Kalińska from the Faculty of Economic Sciences and Management has been appointed Deputy Rector for the safety of students and doctoral students. Her responsibilities include: taking care of and ensuring the sense of safety of students and doctoral students by recognising all kinds of risks, reacting to them, and organising substantive assistance to people whose safety has been compromised.
Dr Bartłomiej Chludziński from the Faculty of Law and Administration became an university ombudsman. His responsibility is to resolve disputes between students, doctoral students and staff of the Nicolaus Copernicus University, in particular through mediation. Information on the regulations in force at the university can also be obtained in the ombudsman's office. The ombudsman's activities are guided by the principles of confidentiality, impartiality and neutrality.
Order of the Rector - 19 October
Welcome foreign students!


Any luck with finding a mouse?
Do you REALLY whant 2 get into THIS?
Any luck with the mice?
Why is it allways me?
Now you just wait till I am old enough to rule my world!
Ya Ya: Who is aproching my beach?
Californian girls and the beachboy?


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Trending Now in Food
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Ben's Chili Bowl Owner Virginia Ali Turns 87 Today. Here's What She's Most Grateful For.
"I'm still able to eat anything I want."
Written by Jessica Sidman
| Published on December 17, 2020
Virginia Ali then and now. Photographs courtesy Ben's Chili Bowl.
Ben's Chili Bowl owner Virginia Ali—as much a local treasure as the restaurant she founded with her husband Ben Ali in 1958—turns 87 today. Earlier this fall, we asked Ali what she was most grateful for amid such a difficult year.
What's this pandemic been like for Ben's as a business and you personally?
It's certainly been a challenge for the business, of course, as it has for all businesses and particularly small ones. It's unlike anything I've experienced in my 87 years. We're doing our best day by day. The love and support that we've gotten from our community has been absolutely heartwarming.
Because I am almost 87, I've been asked not to hang out there as much as I normally do. It's just a pleasure for me to meet and greet our guests and see our team members. It keeps me healthy and I miss being able to do that, although I sneak into it way too often. I stayed at home the first month and I thought, 'OK, this is not healthy for me.' My energy comes from the Chili Bowl. When I walk in the door and I meet these wonderful people from all walks of life and from all over the world, I am just energized. But now I'm not able to do that so much. There's large tour groups and people from all over the country and our neighbors and everything. We want everybody to be safe and to be healthy.
How is Ben's doing?
We just have to stay there day in and day out as a family and serve our guests. We're going to keep it going. We're going to do our very best to keep it going. It's difficult because we had these hours 7 AM to 3 AM and 4 AM on weekends. Well, that's all gone. So we're taking it one day at a time.
What do your kids say about you going into the store and your own personal risk?
They've made sure that I've got all the sanitation I need. I've got my mask and all kinds of things. And so I promised to protect myself with a mask at all times. And hand sanitizers in my car and my house and my office and everywhere I go. They would like me not to mingle too much with large crowds, and so I don't. I do the one-on-one and keep the social distancing there going, and I think that keeps you healthy, I really do.
Are there specific gestures that you've been particularly grateful for?
The very warm letters of support and encouragement—and many with checks. We take those checks and turn them into giving back to our community by providing lunches for the medical staff at Howard University Hospital or the Washington Hospital Center and the fire department, some teachers, and even the [Black Lives Matter] demonstrators.
How many letters have you received?
Oh my gosh, we've received quite a few letters of encouragement. We've been into the community for 62 years, so we've got three or four generations. We've got the older folks that are calling and saying, 'Ben's cannot go anyplace." And then the younger folks that have come to school here, just generations of them from out of town and locally, that say Ben's was a part of their college experience or Ben's was a part of their childhood experience and 'we need you to stay in place and included is a check for $25 or $500 or whatever.'
Do you have any indulgences or daily rituals that you're grateful for?
I've got a great appetite, and I'm still able to eat anything I want. I can have my beef chili dogs anytime I want, and I love our chocolate shakes, and I'm still able to have that. I did have the beef chili dog every day for about 30 years. I don't do it every single day anymore, but I have it as often as I like. I have a chocolate milkshake maybe about three times a week.
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More: Ben's Chili BowlGratitudeVirginia Ali
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Jessica Sidman covers the people and trends behind D.C.'s food and drink scene. Before joining Washingtonian in July 2016, she was Food Editor and Young & Hungry columnist at Washington City Paper. She is a Colorado native and University of Pennsylvania grad.
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Home » Lifestyle » 'Queer Eye' Seasons 4 & 5 Are Officially Happening & The Fab 5 Are Leaving The Midwest
'Queer Eye' Seasons 4 & 5 Are Officially Happening & The Fab 5 Are Leaving The Midwest
06/18/2019 Comments Off on 'Queer Eye' Seasons 4 & 5 Are Officially Happening & The Fab 5 Are Leaving The Midwest
Queer Eye's Fab Five is taking on a whole new meaning now, because the hit Netflix makeover series just got renewed through Season 5. The streaming service announced that the feel-good juggernaut will be returning for not just one, but two more seasons… and the next new season is going to premiere in just one month! And that is not the only surprise these two new seasons have in store. Now that we know Queer Eye Seasons 4 and 5 are happening, here is what fans can expect from the two new seasons.
Fans only have to wait one month for the new season of Queer Eye to hit Netflix, because the network revealed that it will be premiering Season 4 on Friday, July 19. The new season will be set in Kansas City, Missouri, the same place as Season 3. But surprisingly, the also-picked-up fifth season of the show will move the Fab Five to the East Coast for the first time, as it will be set in Philadelphia, Pennsylvania. The location choices seem to be setting up a pattern of Queer Eye filming two seasons per city before traveling to a new town. Philadelphia is also a particularly interesting destination since Fab Five culture expert Karamo Brown first got his start in reality television in that city as a cast member on 2004's The Real World: Philadelphia. The Fab Five are set to begin filming their Philly-set fifth season next month, and it will air on Netflix sometime in 2020.
The Queer Eye cast announced the two new seasons in a short video that revealed the Season 4 premiere date next month and hinted at Season 5 being set in Philadelphia with a short cameo from the Philadelphia Flyers' viral mascot Gritty.
Although Netflix has not yet revealed how many episodes each new season will contain, it is most likely that both Seasons 4 and 5 will consist of eight episodes, the same episode count as the first three seasons.
And these two new seasons are not the only things that Queer Eye fans have to look forward to in the near future. Sometime after Season 4 premieres, Netflix will also drop a new Queer Eye special in which the Fab Five travel to Japan to help out some people there. Queer Eye: We're in Japan! will consist of four episodes, and it will be premiering later this year.
It is hard to believe that Jonathan Van Ness, Tan France, Karamo Brown, Bobby Berk, and Antoni Porowski have only been household names for a year at this point. The Queer Eye for the Straight Guy reboot's new Fab Five quickly shot to fame shortly after the first season of the show aired in early 2018. Since then, the guys have starred in two subsequent seasons of the hit show. Most recently, they proved their pop culture cachet by appearing in Taylor Swift's latest music video, "You Need to Calm Down."
The fourth season of Queer Eye will premiere on Netflix on Friday, July 19.
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We've been enjoying our Tower Garden for 3 years now, but we just got our brand new LED grow lights for this winter, and I have fallen in love all over again!
Before owning our Tower Garden, we tried a couple dirt gardens without success. The time and maintenance a traditional garden requires just isn't in our lifestyle. When I learned about the Tower, and how it is dirt free, it waters itself, yields 30% more, three times faster, while only using 2% of the water of a traditional garden, I bought one immediately!
Not only have we had success with our Tower Garden, but we can hardly keep up with all the produce! We have had our Tower both outdoor and indoor, but MY favorite way to grow is indoor. I love that with the grow lights, the Tower Garden can work for anyone! Urban studios, apartments, college dorms, rooftops, classrooms, restaurants…people who don't have the space for a traditional garden, can still grow their own food!
The original lights were great, but they did get hot, so if my plants got really big, the edges would get burnt. With the new LED lights, they don't get hot, so no burning. The older lights were attached to a tomato cage and were fixed. These new lights are on moveable arms, allowing us to move them closer or farther from the plants as needed.
We are 12 days in with these new lights and new, baby seedlings, and we are already ready to harvest some of our lettuce!


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15 Strangest Things We Discovered Under Frozen Ice In Antarctica
by Jacob Ormrod
in Pop Culture
Human beings often feel as if they've been around for a long time, but the planet has definitely been here for a lot longer. Despite the fact that we have spent a lot of effort trying to map everything out, there are still a lot of places we haven't been able to explore as much as we would like.
One of the places on the planet that continues to be almost impossible to inhabit, thereby continuing to surprise us, is Antarctica. We wanted to take a look at some of the truly surprising things that have been found frozen under the ice of Antarctica, so we've pulled together some of the weirdest discoveries out there.
It's about time we got started. It's time to dive right in and take a look at some of the strangest things ever found under the ice. We cannot believe that the things on this list are real!
15 Mountain Lakes
via: nationalgeographic.com
Using an echo machine, scientists were able to discover the first of the under-ice lakes in the 1970s.
Since then, the technology has got better and they've managed to discover more and more, some of the taking up enough space to include mountain ranges, all contained beneath the ice.
14 See-Through Fish
via: sciencefriday.com
One of the theories for why animals like this don't look like those on the ground is that they've never had to look appealing to other animals, unlike the animals we see every single day.
It's believed that the reason it can be seen through is that it doesn't have the protein hemoglobin, which is what turns our blood red.
13 Gamburtsev Mountain Range
via: foxnews.com
Despite how impressive science may be in the modern world, there are some things that it just can't answer, including how this mountain range still manages to exist under a thin sheet of ice.
By all accounts, it should have been gone well before human beings made their way there!
via: bioweb.ie
Pretty much everything about Antarctica has been a surprise to discover. While nobody thinks of plants when they think about the icy area, scientists have found various types of lichen and moss under the ice that are able to survive extremely cold temperatures.
Who knows what else we'll find under the ice one day!
11 Rainforests
via: comicvine.com
We may all think about Antarctica a certain way, like a huge white abyss that stretches on for miles and miles. However, it wasn't always like this!
Scientists have discovered fossilized wood and the impressions of old leaves that show there was once ancient rainforests where there is now snow and ice.
10 German Base?
via: history.com
When it comes to theories about the Germans, it's smart to be as skeptical as possible. However, it is true that Hitler was exploring Antarctica and did build a hidden base just 960km from the North Pole.
Thankfully, the crew there was poisoned by tainted meat and they ended up abandoning it.
9 Oil
via: antractica.gov.au
If modern history has taught us anything, it's that human beings will do anything to get their hands on oil, no matter where it is.
However, many people don't realize that there is oil under Antarctica, it's just that many believe it is far too dangerous to start removing it from the Earth. It is illegal to dig for oil there until at least 2048.
8 Methane
via: notes.ru
Nobody is denying that these under ice bubbles look very cool and beautiful, but there's a genuine reason to be scared about them escaping! They're made up of the Methane that is released when microbes digest the various organisms under the ice.
If the ice continues to melt, the methane could be released, which could be yet more dangerous for the environment.
7 Dangerous Bacteria
via: thetravel.com
There is so much we don't yet fully understand what lurks in Antarctica. Some scientists have managed to find some microbes that date all the way back to 420,000 years ago.
Some believe that these microbes could even be dangerous, so let's hope that they're never released.
6 Elongated Skulls
via: science.com
It should come as no surprise that many people see these skulls as proof of some sort of otherworldly life, but it's actually much more bizarre than that.
Yes, human beings in the past used to purposefully do this to their own heads! It has been found in other areas of the world as well, not just in Antarctica.
5 An Unknown Creature
via: nationalgeographic.org
We often think that most creatures out there have now been discovered, but places like Antarctica are constantly proving us wrong.
This tiny octopus is a relatively new discovery, and scientists have been finding out new things about them within the last decade! We can't be the only people who think that it's very cute?!
4 Some Very Old Whiskey
via: thehistoryblog.com
There are some people out there that would do anything to get their hands on some good scotch that's over a hundred years old. However, the scientists waited years to make sure that they had the right tools to remove this stuff from the ice without breaking it.
We can only imagine they had quite the party with the bottles that survived!
3 Petrified Animal Remains
via: hearstapps.com
Scientists once found a distant relative of what we now call mammals under the ice of Antarctica. Believed to be over 250 millions years old, the cold had petrified the animal remains and have given scientists a chance to truly learn from them.
The size of a cat, this distant relative also laid eggs.
2 Ancient Fossils
via: gizmodo.com
Since the 1980s, scientists have been finding more and more ancient fossils, most of them belonging to prehistoric birds and marine life.
Over 71 million years old, these fossils are genuinely amazing finds that have been waiting under the ice for someone to come along and catalog them.
1 Meteorites
via: nasa.gov
There are a lot of people out there that will travel to Antarctica in search of meteorites, and that's because so many of them end up landing there.
Once they do, they will often stay above the surface, but some have been found to crash through the ice and wait underneath for somebody to discover them.
Sources - NASA, Gizmodo & History Blog
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