# Amherst 2012

The next workshop will take place in Amherst on March 31- April 1, with a special lecture on Friday afternoon, March 30.

**Organizers: **Paul Hacking, Jessica Sidman, and Jenia Tevelev

Funded by NSF, University of Massachusetts, Amherst College, and Mt Holyoke College

**Registration is now closed.**

**Speakers:**

Kai Behrend (University of British Columbia)

Maksym Fedorchuk (Columbia University)

Alexander Goncharov (Yale University)

Brendan Hassett (Rice University)

Klaus Hulek (Leibniz Universität Hannover)

Bernd Sturmfels (University of California at Berkeley)

Bianca Viray (Brown University)

Claire Voisin (Institut de Mathématiques de Jussieu)

**Videos of Lectures**

Maksym Fedorchuk: Finite Hilbert stability of (bi)canonical curves

Brendan Hassett. Pre-talk. Talk: Families of quartic del Pezzo surfaces

Klaus Hulek: The locus of intermediate Jacobians of cubic threefolds

Bernd Sturmfels. Pre-talk. Talk: Non-negative polynomials versus sums of squares

Bianca Viray: Transcendental Brauer elements via descent on elliptic surfaces

Kai Behrend: Differential graded schemes via curved differential graded Lie algebras

Claire Voisin. Pretalk. Talk: Integral Hodge classes and birational invariants

Radu Laza and Eyal Markman: open problem session

Alexander Goncharov: Ideal webs and moduli spaces of local systems on surfaces

**Schedule**

All lectures and activities will be held in the Integrated Science Building (ISB 135 Auditorium) except for the poster session, wine and cheese reception, and the conference dinner, which will be held in the Campus Center

Friday March 30

3:30-4:30

4:30-6:00

Registration and coffee

**Maksym Fedorchuk: Finite Hilbert stability of (bi)canonical curves**

Saturday March 31

Sunday April 1

**Abstracts:**

**Differential graded schemes via curved differential graded Lie algebras. **Kai Behrend (University of British Columbia)

I will explain how bundles of curved differential graded Lie algebras give rise to a particularly simple class of derived schemes, and describe two examples: derived moduli of stable sheaves on a projective variety, and derived moduli of stable non-commutative projective schemes.

**Finite Hilbert stability of (bi)canonical curves.** Maksym Fedorchuk (Columbia University)

We will discuss stability of finite Hilbert points of the general (bi)canonical curve of arbitrary genus (joint work with Jarod Alper and David Smyth). The generic stability result opens the door to analyzing a whole menagerie of new GIT quotients, which are expected to be log canonical models of the moduli space of stable curves. In low genus this expectation can be verified, and a complete GIT analysis is possible. We will discuss an instance of such an analysis leading to a functorial description of the final log canonical model of the moduli space of stable curves of genus 5 (joint work with David Smyth).

**Ideal webs and moduli spaces of local systems on surfaces.** Alexander Goncharov (Yale University)

A decorated surface is a topological surface (with or without boundary) with a finite collection of points on it, considered modulo isotopy. We introduce ideal webs on a decorated surface S, and show that an ideal web gives rise to a coordinate system of (framed) SL_n-local systems on S.

**Families of quartic del Pezzo surfaces. **Brendan Hassett (Rice University)

We discuss the structure and classification of families of quartic del Pezzo surfaces over the projective line. We focus on the geometry of their spaces of sections and their interaction with the intermediate Jacobian of the total space. Our motivation comes from arithmetic questions for function fields of curves over finite fields. (joint with Tschinkel)

**The locus of intermediate Jacobians of cubic threefolds.**** **Klaus Hulek (Leibniz Universität Hannover)

The locus of intermediate Jacobians of cubic threefolds forms a 10-dimensional cycle in the moduli space A_5 of ppav of dimension 5. We shall discuss the class of this cycle and the intersection of its closure with the boundary. We will also discuss some natural higher-dimensional generalizations. This is joint work with Sam Grushevsky.

**Non-negative polynomials versus sums of squares.** Bernd Sturmfels (UC Berkeley).

We discuss the geometry underlying the difference between non-negative polynomials and sums of squares. The hypersurfaces that discriminate these two cones for ternary sextics and quaternary quartics are shown to be Noether-Lefschetz loci of K3 surfaces. The projective duals of these hypersurfaces are defined by rank constraints on Hankel matrices. We compute their degrees using numerical algebraic geometry, thereby verifying results due to Maulik and Pandharipande. The non-SOS extreme rays of the two cones of non-negative forms are parametrized respectively by the Severi variety of plane rational sextics and by the variety of quartic symmetroids. This lecture is based on work of Greg Blekherman, and on our joint paper with Jonathan Hauenstein, John Christian Ottem and Kristian Ranestad.

**Transcendental Brauer elements via descent on elliptic surfaces.** Bianca Viray (Brown University)

Transcendental elements in the Brauer group are notoriously difficult to compute. Wittenberg and Ieronymou have worked out explicit representatives for 2-torsion elements of elliptic surfaces, in the case that the Jacobian fibration has rational 2-torsion. We use ideas from descent to develop techniques to study the 2-torsion elements of elliptic surfaces without an assumption on the 2-torsion.

**Integral Hodge classes and birational invariants. **Claire Voisin (Institut de Mathématiques de Jussieu).

The group of integral Hodge classes of degree 2i on X (a smooth projective complex variety) modulo the group of cycle classes is a birational invariant for i=4 and i=n-1, n=dim X. We will discuss the vanishing and nonvanishing of these groups for rationally connected varieties. ** **

**Poster Session**

5:30-7:00 pm, Campus Center

**Kulikov surfaces **Stephen Coughlan (UMass Amherst) Download poster

Kulikov surfaces are a nice example of surfaces of general type with p_g=0 and K^2=6. We explain a couple of different ways to construct such surfaces, and prove that they form a connected component of the moduli space. This is joint work with Tz On Mario Chan.

**Snc Fano varieties and log Fano manifolds** Kento Fujita (RIMS and Princeton University) Download poster

Snc (simple normal crossing) Fano varieties and log Fano manifolds are one of the most natural generalizations of Fano manifolds. In this poster, I classified those objects with large indices, which are related to Mukai conjecture.

**Degenerations of surfaces and vector bundles** Anna Kazanova (UMass Amherst) Download poster

A recent construction of Hacking relates the classification of stable vector bundles on a surface of general type with p_g=0 and the boundary of the moduli space of deformations of the surface. We analyze this correspondence for rank 2 and K^2=1.

**Integration on GIT quotients by non-abelian groups** Zachary Maddock (Columbia University)** **Download poster

For a reductive group G acting linearly on a projective variety, let the ratio R to be the integral on the GIT quotient X//G of a dimension 0 Chow class \alpha divided by the integral on the quotient by the maximal torus, X//T, of a lift of \alpha capped with T-equivariant Chern class of the representation Lie(G)/Lie(T). I show that this integral is independent of the choice of maximal torus, the choice of linearization, and the choice of projective variety. I give an algebraic proof when G is of type A_n, that R = (n+1)! = |W|, the order of the Weyl group. This ratio R is related to formulas in symplectic geometry in the works of L. Jeffreys, F. Kirwan, and S. Martin. Using Martin's result, I can show that this implies that R = |W| for any reductive group. Providing a purely algebraic proof of this result is still an open area of research.

**Cohomology ring of the Jacobi Factor of quasi-homogeneous plane curve singularities** Mikhail Mazin (Stony Brook) Download poster

We study the cohomology ring of the Jacobi Factor of quasi-homogeneous plane curve singularities and compare it with a ring defined by L. Goettsche, B. Fantechi, D. van Straten. In some cases we construct an explicit bijection between a basis in the ring and the cells of the Jacobi Factor. This is a joint work in progress with Eugene Gorsky.

**Double MV Cycles and Affine Crystal Combinatorics **Dinakar Muthiah (Brown University) Download poster

In this poster, I briefly review some facts about the Geometric Satake Correspondence and MV (Mirkovic-Vilonen) cycles in the Affine Grassmannian. Studying MV cycles naturally leads to MV polytopes and surprising connections to total positivity and PBW bases.

Then I introduce Double MV cycles, the generalization of MV cycles to affine groups (loop groups). I discuss a connection with the Naito-Sagaki-Saito crystal in type A. Finally, I explain some conjectural connections with Affine MV Polytopes and related works of various authors.

**Boundary Divisors in the Moduli Space of Stable Quintic Surfaces** Julie Rana (University of Massachusetts Amherst) Download poster

The moduli space of curves of genus g has a compactification, the moduli space of stable curves. A stable curve is a connected curve with at most nodal singularities and finite automorphism group. The moduli space of minimal surfaces of general type with fixed invariants admits an analogous compactification, the moduli space of stable surfaces, introduced by Kollár-Shepherd-Barron and Alexeev. Here, stable surfaces are connected projective surfaces with ample canonical class and semi log canonical singularities. There are some expected divisors in the boundary of this moduli space. We describe one such divisor in the moduli space of stable quintic surfaces, the divisor corresponding to surfaces with Wahl singularities. A different set of expected boundary divisors corresponds to weighted blowups of quintic surfaces with Fuchsian singularities. We discuss classification of these surfaces.

**Apolarity for determinants and permanents of generic matrices **Sepideh Shafiei (Northeastern) Download poster

We show that the annihilator ideal (in the sense of the apolar pairing; i.e., Macaulay's inverse system) of the determinant and the permanent of a generic matrix, the annihilator ideal of the Pfaffian of a generic skew symmetric matrix and the annihilator ideal of the Hafnian of generic symmetric matrix are generated in degree two. As a consequence we give a lower bound for the cactus rank and rank of the determinant and the permanent using a result of K. Ranestad and F.O. Schreyer.

**The Schottky problem in genus 5 **Charles Siegel (University of Pennsylvania) Download poster

This poster will describe a solution to the Schottky problem for abelian varieties of dimension five, exploiting incidence relations of the fibers of the Prym map and a related map defined by theta functions.

**Equations of Riemann surfaces with automorphisms** David Swinarski (Fordham University) Download poster

We present a pseudoalgorithm for finding equations of Riemann surfaces with suitably large automorphism groups. The method is a combination of the Eichler trace formula and the projection formula from representation theory, and has been used to produce equations of genus 4, 5, and 6 curves with automorphism groups satisfying |G| > 4(g-1).

**Partial compactification of the zero section of the universal abelian variety** Dmitry Zakharov (Stony Brook University) Download poster

The moduli space of principally polarized abelian varieties is one of the central objects of study in algebraic geometry. The moduli space is not compact, and admits several natural compactifications. All of these compactifications are extensions of Mumford's partial compactification of semiabelic varieties of torus rank one. The partial compactification is the base for a universal family that admits a zero-section. In our joint work with Sam Grushevsky, we calculate the class of the zero section in the Chow ring of the partial compactification of the universal abelian variety.

**Modular Forms and Special Cubic Fourfolds** Letao Zhang (Rice University) Download poster

We study the degree of the special cubic fourfolds in the Hilbert scheme of cubic fourfolds via a computation of the generating series of Heegner divisors of even lattice of signature (2, 20). (Joint work with Z. Li)

**Parking options:** (1) Campus parking garage (2) After 5pm Friday: ISB parking lot, or most other UMass parking lots (3) Before 5pm Friday: Parking meters (on N Pleasant next to roundabout).

See the map above or the UMass parking map here for more details.

Note: Participants staying in the UMass hotel will receive a parking pass to the campus parking garage (one per room).

**Participants**

Anno, Rina (University of Massachusetts, Amherst)

Arap, Maxim (Johns Hopkins University )

Ascher, Ken (Stony Brook)

Atanasov, Atanas (Harvard University)

Atyam, Anant (Stony Brook)

Bakker, Benjamin (Courant Institute)

Banerjee, Soumya (Yale University)

Behrend, Kai (University of British Columbia)

Belcastro, Sarah-Marie (UMass Amherst)

Bourdon, Abbey (Wesleyan University)

Braden, Tom (UMass Amherst)

Bujokas, Gabriel (Harvard)

Burr, Michael (Fordham University)

Castravet, Ana-Maria (Ohio State University)

Cattani, Eduardo (UMass Amherst)

Chen, Dawei (Boston College)

Chen, Qile (Columbia University)

Cooper, Yaim (Princeton)

Coughlan, Stephen (UMass)

Cramer, Tim (Yale)

Cox, David (Amherst College)

De Cataldo, Mark (Stony Brook)

Deopurkar, Anand (Harvard University)

Di Brino, Gennaro (Yale)

Di Cerbo, Gabriele (Princeton University)

Dionne, Chris (Queen's university)

Dumitrescu, Olivia (Univerity of Davis, California)

Dwivedi, Shashank (MIT)

Dykas, Nathan (University of Connecticut)

Sommers, Eric (UMass Amherst)

Fedorchuk, Maksym (Columbia University)

Fintzen, Jessica (Harvard University)

Foster, Tyler (Yale)

Friedlander, Holley (UMass Amherst)

Friedmann, Tamar (University of Rochester)

Fujita, Kento (RIMS and Princeton University)

Gallardo, Patricio (Stony Brook)

Garcia-Raboso, Alberto (University of Pennsylvania)

Gassert, Alden (UMass)

Gillam, Danny (Brown University)

Giovanni Faonte (Yale)

Goncharov, Alexander (Yale University)

Gonzalez, Jose (University of British Columbia)

Gorskiy, Evgeny (Stony Brook University)

Grieve, Nathan (Queen's University)

Grushevsky, Samuel (Stony Brook University)

Gutt, Jan (Stony Brook University)

Hacking, Paul (UMass)

Harper, John (University of Massachusetts Amherst)

Hassett, Brendan (Rice University)

Hasson, Hilaf (University of Pennsylvania)

Hatley, Jeffrey (UMass)

Havens, Andrew (UMass)

Hering, Milena (University of Connecticut)

Ho, Wei (Columbia University)

Hulek, Klaus (Leibniz Universitaet Hannover)

Humphreys, Jim (U. Mass. Amherst)

Iarrobino, Anthony (Northeastern University)

Jensen, David (Stony Brook University)

Kamenova, Ljudmila (Stony Brook University)

Kang, Su-Jeong (Providence College)

Karzhemanov, Ilya (NYU)

Kazanova, Anna (UMass Amherst)

Kiluk, Andrew (Columbia University)

Kleiman, Steven (MIT)

Knecht, Amanda (Villanova University)

Knight, Fu (Rutgers University)

Laza, Radu (Stony Brook University)

Lazarsfeld, Robert (U Mich)

Lesieutre, John (MIT)

Li, Zhiyuan (Rice University)

Lin, Yinbang (Northeastern University)

Little, John (College of the Holy Cross)

Liu, Yu-Han (Princeton University)

Liu, Tiankai (Massachusetts Institute of Technology)

Lopez Martin, Alberto (Tufts University)

Lu, Zhentao (UPenn)

Macias Marques, Pedro (Universidade de Evora /Northeastern University)

Maddock, Zachary (Columbia University)

Markman, Eyal (University of Massachusetts at Amherst)

Mayanskiy, Evgeny (The Pennsylvania State University)

Mazin, Mikhail (Stony Brook)

Miasnikov, Nikita (GC, CUNY)

Muthiah, Dinakar (Brown University)

Oblomkov, Alexei (University of Massachusetts)

Oloo, Stephen (UMass Amherst)

Ou, Tze-Chun (UConn)

Pal, Vivek (Columbia)

Papantonopoulou, Aigli (The College of New Jersey)

Patakfalvi, Zsolt (Princeton University)

Patel, Anand (Harvard University)

Pecharich, Jeremy (Mt. Holyoke)

Perry, Alex (Harvard)

Pinkham, Henry (Columbia University)

Potashnik, Natasha (Columbia University)

Rana, Julie (University of Massachusetts Amherst)

Riedl, Eric (Harvard)

Rizzardo, Alice (Columbia University)

Ross, Joe (University of Southern California)

Routis, Evangelos (Brown University)

Sacca', Giulia (Princeton University)

Saltman, David J (Center for Communications Research )

Samouil Molcho (Brown )

Schiffler, Ralf (University of Connecticut)

Shafiei, Sepideh (Northeastern)

Shen, Linhui (Yale University)

Shenfeld, Daniel (Princeton University)

Shor, Caleb (Western New England University)

Sidman, Jessica (Mount Holyoke)

Siegel, Charles (University of Pennsylvania)

Smirnov, Ilia (Queen's University)

Smyth, David (Harvard University)

Staal, Andrew (Queen's University)

Starr, Jason (Stony Brook University)

Stout, Andrew (CUNY, Graduate Center)

Sturmfels, Bernd (UC Berkeley)

Mehrotra, Sukhendu (University of Wisconsin Madison)

Svaldi, Roberto (MIT)

Swinarski, David (Fordham University)

Tanimoto, Sho (Courant Institute, NYU)

Teixidor, Montserrat (Tufts University)

Tevelev, Jenia (UMass)

Thaddeus, Michael (Columbia University)

Tian, Zhiyu (Caltech)

Tramel, Rebecca (University of Connecticut)

Tu, Yu-Chao (Princeton)

Tzeng, Yu-jong (Harvard University)

Ulirsch, Martin (Brown University)

Urzua, Giancarlo (Pontificia Universidad Catolica de Chile)

Van Steirteghem, Bart (Medgar Evers College - CUNY

Viray, Bianca (Brown University)

Voisin, Claire (Institut de Mathematiques de Jussieu)

Wang, Bin (Rhode Island College)

Wen, Jun (Stony Brook University)

Willocks, Bradley (UMass Amherst)

Wilson, Tobias (UMass)

Woolf, Matthew (Harvard)

Xhumari, Sandi (UConn)

Xia, Jie (Columbia University)

Xuanyu, Pan (Columbia University)

Yang, Yanhong (Columbia University)

Yang, Yaping (Northeastern)

Zakharov, Dmitry (Stony Brook University)

Zhang, Dingxin (Stony Brook University)

Zhang, Letao (Rice University)

Zhang, Zheng (Stony Brook University)

Zhang, Zili (Stony Brook University)

Zhao, Gufang (Northeastern)

Zhu, Yi (Stony Brook University)

Zong, Runpu (Princeton University)

**Workshop poster**: available here

**About AGNES**

AGNES is a new series of weekend workshops in algebraic geometry. One of our goals is to introduce graduate students to a broad spectrum of current research in algebraic geometry. AGNES is held twice a year at the participating universities in the Northeast.

**Scientific Committee:**

Dan Abramovich (Brown), Joe Harris (Harvard), Aise Johan de Jong (Columbia),

Mikhail Kapranov (Yale), János Kollár (Princeton), James McKernan (MIT)

**Organizing committee:**

Dan Abramovich (Brown), Arend Bayer (UConn), Alexander Braverman (Brown),

Sam Grushevsky (Stony Brook), Paul Hacking (UMass Amherst), Milena Hering (UConn),

Mikhail Kapranov (Yale), Radu Laza (Stony Brook), James McKernan (MIT),

Sam Payne (Yale), Jason Starr (Stony Brook), Jenia Tevelev (UMass Amherst).