AGNES Workshop

Algebraic Geometry

Northeastern Series

Saturday and Sunday
October 31st and November 1st
Special lecture on Friday afternoon, October 30th

Stony Brook University Mathematics Department, Room S–240 

This workshop is funded by the National Science Foundation.

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AGNES Workshop Series

AGNES, short for Algebraic Geometry Northeastern Series, is a new series of weekend workshops in algebraic geometry to be held at Stony Brook University, UMass Amherst, and University of Connecticut. Although all are welcome, the emphasis is on young mathematicians: graduate students and post-docs. Funding is available to help support graduate students and some post-docs.

Graduate students and post-docs applying for financial support should ask their advisor or post-doctoral mentor to send a brief recommendation to one of the AGNES organizers.

The last day to apply for financial support is Thursday, October 22nd. After that date, additional participants will have to make their own hotel arrangements and pay for their own travel.

Focused Research Group

This first AGNES workshop is also part of the Focused Research Group on "Geometry of moduli spaces of rational curves with applications to Diophantine problems over function fields" organized by A. Johan de Jong, Brendan Hassett, Jason Starr and Yuri Tschinkel. So there will be lectures focused on this topic.

Speakers

• Arend Bayer (University of Connecticut)

• Ana–Maria Castravet ( University of Arizona)

• Herb Clemens ( The Ohio State University)

• William Fulton ( University of Michigan)

• Amanda Knecht ( University of Michigan)

• Li Li ( University of Illinois at Urbana–Champaign)

• Mike Roth (Queen's University)

• Jenia Tevelev ( University of Massachusetts Amherst)

Scientific committee

Dan Abramovich (Brown), Aise Johan de Jong (Columbia), Joe Harris (Harvard), Mikhail Kapranov (Yale), János Kollár (Princeton), James M^cKernan (MIT).

AGNES organizers

Arend Bayer, Samuel Grushevsky, Paul Hacking, Milena Hering, Jason Starr, Jenia Tevelev.

Local Stony Brook organizers

Mark Andrea de Cataldo, Matt DeLand, Samuel Grushevsky, Ljudmila Kamenova, Alexander Kirillov, Jr., Radu Laza, Jason Starr, Andrew Young, Aleksey Zinger.

Hotel information

Participants who have been offered accomodation will be staying at the Holiday Inn Express in Stony Brook. Please contact Jason Starr and Nancy Rohring as soon as possible to confirm your arrival date and departure date. For participants who are making their own arrangements, please still contact us to keep us informed of your plans. The hotel does have a shuttle service to pick up and drop off guests at the Stony Brook LIRR station, the Port Jefferson ferry, and the Islip MacArthur airport.

Local information

• Here are some directions on getting to the mathematics department. For visitors who are flying in, most visitors find it convenient to fly in to JFK or LGA (LaGuardia) and then transfer to the Long Island Rail Road at Jamaica station. There is an AirTrain connecting JFK to the Jamaica LIRR station, but it costs $5 to "escape" from the AirTrain. From LGA it is simplest to take a taxi to the Jamaica LIRR station (although there are many other options for the adventurous). For travellers who intend to use the LIRR for at least part of their journey, please see the next item.
• For visitors travelling through New York City, here is the schedule for the Long Island Rail Road. The Stony Brook station is on campus, a few minutes walk from the mathematics department. Some visitors may prefer to arrive at the Ronkonkoma LIRR station and take a taxi to campus or your hotel.
• Here is an online map of Stony Brook's campus. The LIRR station is in B2 and B3 on the map. To get to the math department, walk along Circle Road south to Gym Road in the NW corner of B4, then walk along Gym Road east to the gym parking lot on the E edge of B4. Then cross John S. Toll road, proceed up the concrete stairs, and continue to the Math Tower at the S edge of C4 (about a 7 minute walk total). A webpage with more maps, including Google maps, is here.
• For participants who are driving, please contact Jason Starr and Nancy Rohring about getting a visitor's parking permit for campus parking lots.
• Aleksey Zinger has a nice list of on-campus and off-campus dining options here.

Wireless internet access

Non-Stony Brook participants who would like wireless access should contact Pat Tonra in advance and let him know all of the following information.

• First Name
• Last Name
• E-mail
• Phone Number
• Institution

Workshop dinner

The workshop dinner will be on Saturday night at John Harvard's Brew House in Lake Grove. The entree will be a choice of salmon, steak or pasta. Participants with dietary restrictions should contact Jason Starr and Nancy Rohring in advance so we can make sure you are accommodated.

The cost of dinner will be covered for all out–of–town participants and organizers. The size of the function room at the restaurant is just large enough to seat all registered out–of–town participants. Other local participants who would like to attend the dinner should contact Nancy Rohring and Jason Starr to determine if that is possible (and to discuss the cost of the dinner).

Alcohol is not included. Participants will have to pay for their own alcohol tab.

Tentative schedule

Special Friday afternoon lecture.

Li Li, Hilbert schemes of points, 4:30 — 5:30 PM

For dinner on Friday, local organizers and participants will take the out–of–town participants to some of the local restaurants.

Saturday and Sunday events.

Saturday	Event	Sunday	Event
8:30AM-9:30AM	Registration and coffee	8:30AM-9:00AM	Coffee
9:30AM-10:30AM	Arend Bayer, The local projective plane, a fractal curve and Γ1(3)	9:00AM-10:00AM	Ana–Maria Castravet, Rational curves of minimal degree on higher Fano manifolds
10:30AM-10:45AM	Break	10:00AM-10:15AM	Break
10:45AM-11:15AM	Pre–lecture	10:15AM-10:45AM	Pre–lecture
11:15AM-12:15PM	Jenia Tevelev, On the cone of effective divisors of M0,n	10:45AM-11:45AM	Herb Clemens, Exploring the Hodge problem
12:15PM-1:45PM	Lunch	11:45AM-12NOON	Break
1:45PM-2:15PM	Pre–lecture	12NOON-1:00PM	Amanda Knecht, Rationally connected varieties over ℚpnr
2:15PM-3:15PM	Mike Roth, A local-global principle for weak approximation of varieties over function fields	1PM	Workshop ends
3:15PM-3:45PM	Break
3:45PM-4:15PM	Pre–lecture
4:15PM-5:15PM	William Fulton, Character formulas
7PM-10PM	Dinner (John Harvard's)

• Arend Bayer, The local projective plane, a fractal curve and Γ₁(3) .

  I will report on joint work with E. Macri on the space of stability conditions for the derived category of the total space of the canonical bundle on the projective plane. It is a 3–dimensional manifold, with many chamber decompositions coming from the behaviour of moduli spaces of stable objects under change of stability conditions.

  I will explain how this space is related to classical results by Drezet and Le Potier on stable vector bundles on the projective plane. Using the space helps to determine the group of auto-equivalences, which includes a subgroup isomorphic to Γ₁(3). Finally, via mirror symmetry, it contains a universal cover of the moduli space of elliptic curves with Γ₁(3)–level structure.

  Hand-written notes by Jason Starr.

• Ana-Maria Castravet, Rational curves of minimal degree on higher Fano manifolds.

  This is joint work with Carolina Araujo. We will discuss a notion of higher Fano variety introduced by de Jong and Starr. We will especially study what one can say about the families of minimal degree rational curves that sweep out such a higher Fano variety. Related questions: Can one classify higher Fanos? Is there an inductive structure on the collection of all higher Fano varieties?

  Hand-written notes by Jason Starr.

• Herb Clemens, Exploring the Hodge problem.

  I will give a brief explanation of the Hodge conjecture for complex projective manifolds of even dimension and the use of normal functions in studying it. I will then build on work of Voisin to generalize the notion of normal function in order to create a context for studying the Hodge problem for complex projective manifolds of dimension 2n by induction on n.

• William Fulton, Character formulas.

  In this expository talk, we will give a simple formula, with a simple proof, for the equivariant euler characteristic of an equivariant vector bundle on a complete, smooth variety with an action of a diagonalizable group. On homogeneous varieties this gives Weyl's character formula, and on toric varieties it gives Brion's formula for lattice points in polytopes. This is based on ideas of George Quart in the 1970's and recent conversations with Bill Graham.

  Notes by Paul Hacking.

• Amanda Knecht, Rationally connected varieties over ℚ_p^nr.

  A result of Graber, Harris, and Starr shows that a rationally connected variety defined over the function field of a curve over the complex numbers always has a rational point. Similarly, a separably rationally connected variety over a finite field or the function field of a curve over any algebraically closed field has a rational point. We will discuss the case of rationally connected varieties over the maximally unramified extension of the p-adics. They `usually' have points, and we will define what `usually' means.

  Hand-written notes by Jason Starr.

• Li Li, Hilbert schemes of points.

  I will talk on two problems that are related to Hilbert schemes of points. The first is to study the Hilbert schemes of points on a tame Deligne-Mumford stack. I shall talk on smoothness, connectedness and Betti numbers of such Hilbert schemes (joint work with Jason Starr). The second problem is about generators of the diagonal ideal of (ℂ²)ⁿ and q,t-Catalan numbers (joint work with Kyungyong Lee).

  Li Li's slides

• Mike Roth, A local-global principle for weak approximation of varieties over function fields

  Brendan Hassett and Yuri Tschinkel introduced the study of the "weak approximation problem" for varieties over function fields, motivated by the analogous question over number fields. The question is whether local sections can be approximated to arbitrarily high order by global sections, and thus fits in to a well known (and fruitful) class of "local-global" type questions. The purpose of this talk is to study whether the weak approximation problem is itself purely local. This is joint work with Jason Starr.

  Notes by Paul Hacking.

• Jenia Tevelev, On the cone of effective divisors of M_0,n.

  We give a conjectural description of the cone of effective divisors of the Grothendieck-Knudsen moduli space M_0,n of stable rational curves. Roughly speaking, non-boundary extremal divisors correspond to total stable degenerations (or, equivalently, pair-of-pants decompositions) with some decorations of smooth curves of genus n-3. Joint with Ana-Maria Castravet.

  Hand-written notes by Jason Starr.

  The transparency from the lecture.

Pre–lectures.

Preceding some of the lectures there will be an half-hour discussion to introduce the material discussed in the main lecture.

Workshop participants

• Dan Abramovich, Brown University
• Jarod Alper, Columbia University
• Somnath Basu, Stony Brook University
• Arend Bayer, University of Connecticut
• Soumya Benerjee, Yale University
• Ilke Canakci, University of Connecticut
• Eduardo Cattani, University of Massachusetts Amherst
• Qile Chen, Brown University
• Herb Clemens, The Ohio State University
• Keith Conrad, University of Connecticut
• Peter Dalakov, University of Massachusetts Amherst
• Matt DeLand, Stony Brook University
• Alicia Dickenstein, Universidad de Buenos Aires and MSRI
• Gennaro Di Brino, Yale University
• Colin Diemer, University of Pennsylvania
• Abigail Ebin, Yale University
• Robert Findley, Stony Brook University
• Maksym Fedorchuk, Columbia University
• Conor Frailey, Yale University
• William Fulton, University of Michigan Ann Arbor
• Alberto Garcia-Raboso, University of Pennsylvania
• Noah Giansiricusa, Brown University
• Danny Gillam, Brown University
• Samuel Grushevsky, Stony Brook University
• Weixin Guo, Stony Brook University
• Jan Gutt, Stony Brook University
• Paul Hacking, University of Massachusetts Amherst
• Hilaf Hasson, University of Pennsylvania
• Milena Hering, University of Connecticut
• Wenchuan Hu, IAS
• YoonSuk Hyun, Massachusetts Institute of Technology
• Mikhail Kapranov, Yale University
• Anna Kazanova, University of Massachusetts Amherst
• Leila Khatami, Northeastern University
• Alexander Kirillov, Jr., Stony Brook University
• János Kollár, Princeton University
• Radu Laza, Stony Brook University
• Brian Lehman, Massachusetts Institute of Technology
• Long Li, Stony Brook University
• Li Li, University of Illinois at Urbana–Champaign
• Shuijing Li, Rice University
• Zhiyuan Li, Rice University
• Tsung-Yin Lin, Stony Brook University
• Zachary Maddock, Columbia University
• Eyal Markman, University of Massachusetts Amherst
• Evgeny Mayanskiy, Penn State
• Sukhendu Mehrotra, University of Wisconsin Madison
• Ian Morrison, Fordham University
• Dinakar Muthiah, Brown University
• Alexei Oblomkov, University of Massachusetts Amherst
• Svyatoslav Pimenov, Yale University
• Alexandra Popa, Stony Brook University
• You Qi, Columbia University
• Alice Rizzardo, Columbia University
• Joe Ross, Universität Essen-Duisburg
• Mike Roth, Queen's University
• Ryan Schwarz, University of Connecticut
• Gagan Sekhon, University of Connecticut
• Yijun Shao, University of Arizona
• Mingmin Shen, Columbia University
• Vivek Shende, Princeton University
• David Speyer, Clay Mathematics Institute and Massachusetts Institute of Technology
• Jason Starr, Stony Brook University
• Dennis Sullivan, Stony Brook University and CUNY
• Jingzhou Sun, Johns Hopkins University
• Jenia Tevelev, University of Massachusetts Amherst
• Zhiyu Tian, Stony Brook University
• Gueorgui Todorov, Princeton University
• Giancarlo Urzua, University of Massachusetts Amherst
• Kartik Venkatram, Massachusetts Institute of Technology
• Botong Wang, Purdue University
• Jun Wen, Stony Brook University
• Loy Weng, Stony Brook University
• Jie Xia, Columbia University
• Fei Xu, Rice University
• Chenyang Xu, Massachusetts Institute of Technology
• Pan Xuanyu, Columbia University
• Yanhong Yang, Columbia University
• Yaping Yang, Northeastern University
• Matt Young, Stony Brook University
• Letao Zhang, Rice University
• Yongsheng Zhang, Stony Brook University
• Zheng Zhang, Stony Brook University
• Gufang Zhao, Northeastern University
• Yi Zhu, Stony Brook University
• Zhixian Zhu, University of Michigan Ann Arbor

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Jason Starr
4-108 Math Tower
Department of Mathematics
Stony Brook University
Stony Brook, NY 11794-3651
Phone: 631 632 8270
FAX: 631 632 7631
Jason Starr