Algebraic Geometry at Stanford

(The seminar webpage is here.)

Mihnea Popa (Northwestern University)
September 25, 2015

Positivity for Hodge modules and geometric applications

M. Saito’s theory of Hodge modules provides a powerful generalization of classical Hodge theory that is beginning to find basic applications to birational geometry. One of the main reasons for this is that generalizations of Hodge bundles and de Rham complexes arising in this context satisfy analogues of well-known vanishing and positivity theorems. I will review a number of such results that have been obtained recently, and how they can be applied to deduce new statements regarding holomorphic one-forms and families of varieties of general type. Much of this work is joint with C. Schnell. This talk is intended for a wide algebro-geometric audience, and will contain a review of necessary background from Hodge theory.

Tony Pantev (University of Pennsylvania)
October 2, 2015

Symplectic geometry, foliations, and potentials

I will explain how Lagrangian foliations in derived algebraic symplectic geometry give rise to global potentials. I will also give natural constructions of isotropic foliations on moduli
spaces and will discuss the associated potentials. I will give applications to the moduli of representations of fundamental groups and to non-abelian Hodge theory. This is based on joint works with Calaque, Katzarkov, Toen, Vaquie, and Vezzosi.  This talk is intended for a general algebro-geometric audience, and no advanced background will be assumed.

Alex Perry (Harvard)
October 30, 2015

Categorical joins

Homological projective duality is a powerful theory developed by Kuznetsov for studying the derived categories of varieties. It can be thought of as a categorification of
classical projective duality. I will describe a categorical version of the classical join of two projective varieties, and its relation to homological projective duality. I will discuss some applications to the structure of the derived categories of Fano varieties and to derived equivalences of Calabi-Yau varieties. This is work in progress with Alexander Kuznetsov.

Zijun Zhou (Columbia)
November 13, 2015

Relative orbifold Donaldson-Thomas theory and local gerby curves

In this talk I will introduce the generalization of relative Donaldson-Thomas theory to 3-dimensional smooth Deligne-Mumford stacks. We adopt Jun Li’s construction of expanded pairs and degenerations and prove an orbifold DT degeneration formula. I’ll also talk about the application in the case of local gerby curves, and its relationship to the work of Okounkov-Pandharipande and Maulik-Oblomkov.

Vikraman Balaji (Chennai Mathematical Institute)
November 20, 2015, 2:30-3:30

Degeneration of moduli of Higgs bundles on curves

In this this talk I will discuss the construction of a degeneration of the moduli space of Higgs bundles on smooth curves, as the smooth curve degenerates to a nodal curve with a single node. As an application we get new compactifications of the Picard variety for smooth curves degenerating to irreducible nodal curves (with multiple nodes) which have analytic normal crossing singularities.

Jose Rodriguez (University of Chicago)
November 20, 2015

Numerically computing Galois groups with Bertini.m2

Galois groups are an important part of number theory and algebraic geometry. To a parameterized system of polynomial equations one can associate a Galois group whenever the system has k (finitely many) nonsingular solutions generically. This Galois group is a subgroup of the symmetric group on k symbols. Using random monodromy loops it has already been shown how to compute Galois groups that are the full symmetric group. In this talk, we show how to compute Galois groups that are proper subgroups of the full symmetric group. We conclude with an implementation using Bertini.m2, an interface to the numerical algebraic geometry software Bertini through Macaulay2. This is joint work with Jonathan Hauenstein and Frank Sottile.

Yukinobu Toda (Institute for the Physics and the Mathematics of the Universe, University of Tokyo)
December 4, 2015, 2:30-3:30 (because of the department party)

Non-commutative deformations and Donaldson-Thomas invariants

In this talk, I will show the existence of certain global non-commutative structures on the moduli spaces of stable sheaves on algebraic varieties, whose formal completion at a closed point gives the pro-representable hull of the non-commutative deformation functor of the sheaf developed by Laudal, Eriksen, Segal and Efimov-Lunts-Orlov. I will then introduce the generating series through integrations over Hilbert schemes of points on these NC structures. When the underlying variety is a Calabi-Yau 3-fold, and the moduli space of stable sheaves satisfy some assumptions, this generating series admits a product expansion described by generalized DT invariants. This formula explains the dimension formula of Donovan-Wemyss’s contractions algebras for floppable curves on 3-folds in terms of genus zero Gopakumar-Vafa invariants.

Aaron Landesman (Harvard)
Tuesday, January 5, 2016 in 384-H (Student Algebraic Geometry Seminar)

Interpolation of Projective Varieties

In this talk, we discuss interpolation of projective varieties through points. It is well known that one can find a rational normal curve in through general points. More recently, it was shown that one can always find nonspecial curves through the expected number of general points. We consider the generalization of this question to varieties of all dimensions and explain why rational normal scrolls satisfy interpolation. Time permitting, we’ll also discuss joint work with Anand Patel on interpolation for del Pezzo surfaces and present several interesting open interpolation problems. We’ll place particular emphasis on explaining the standard techniques used to solve interpolation problems: deformation theory, specialization, degeneration, and the gale transform.

Tonghai Yang (Wisconsin)
January 15, 2016, 2:30-3:30 pm (joint with Number Theory)

Generating series of arithmetic divisors on a Shimura variety of unitary type

In this talk we will define a generating series of arithmetic divisors on a Shimura variety of unitary type (n-1, 1), and give a rough idea how to prove that it is modular. This can be viewed as generalization of modular form of weight 2 of Heeger divisors used in Gross-Zagier formula and its generalization Yuan-Zhang-Zhang formula. If time permits, I will mention its application to a special case of Colmez conjecture and `integral’ generalization of the Gross-Zagier and Yuan-Zhang-Zhang formula. This is a joint work with Bruinier, Howard, Kudla, and Rapoport.

Sheldon Katz (UIUC)
January 15, 2016

Stable pair invariants of elliptically fibered Calabi-Yau threefolds and Jacobi forms

I explain the conjectural description of the generating function of stable pair invariants of elliptically fibered Calabi-Yau threefolds with fixed base class and variable fiber class in terms of meromorphic Jacobi forms whose weight grows linearly and whose index grows quadratically with the base class.  This talk is based on joint work with Albrecht Klemm and Minxin Huang.

Jacob Tsimerman (Toronto)
January 22, 2016

Bounding 2-torsion in class groups

(joint with Bhargava, Shankar, Taniguchi, Thorne, and Zhao) Zhang’s conjecture asserts that for fixed positive integers , , the size of the m-torsion in the class group of a degree n number field is smaller than any power of the discriminant. In all but a handful of cases, the best known result towards this conjecture is the ”convex” bound given by the Brauer-Siegel Theorem. We make progress on this conjecture by giving a `”subconvex” bound on the size of the 2-torsion of the class group of a number field in terms of its discriminant, for any value of . The proof is surprisingly elementary, and we give several applications of this result stemming from the case of cubic fields, including improved bounds on the number of fields, and on the number of integer points an elliptic curve can have. Along the way, we prove a surprising result on the shape of the lattice of the ring of integers of a number field. Namely, we show that such a lattice is very limited in how ‘skew’ it can be.

Vivek Shende (Berkeley)
January 29, 2016

Cluster varieties from Legendrian knots

We give a uniform geometric explanation for the existence of cluster structures on the positroid strata, character varieties, wild character varieties, etc.

The point is that such spaces can be identified as the moduli spaces of sheaves with microsupport in an appropriate Legendrian knot. In general, an exact Lagrangian with asymptotics in the knot gives rise to an algebraic torus chart on the moduli space. We construct such fillings for alternating knots, so get charts in correspondence with alternating representatives of the knot in question. Such can be enumerated by bicolored graphs, and we recover the various aspects of the bicolored graph theory — the boundary measurement map of Postnikov; the cluster structure on the positroid varieties, the analogous construction on the character varieties due to Fock and Goncharov — by computing the Floer homology between our fillings.

This talk presents joint work with David Treumann, Harold Williams, and Eric Zaslow.

Izzet Coskun (University of Illinois at Chicago)
April 15, 2016

Birational geometry of moduli spaces of sheaves on surfaces

In this talk, I will discuss recent developments in the birational geometry of moduli spaces of sheaves on surfaces motivated by Bridgeland stability conditions. After reviewing joint work with Arcara, Bertram, Huizenga and Woolf in the case of the projective plane, I will describe how to use Bridgeland stability conditions to construct nef divisors on moduli spaces of sheaves on surfaces. I will illustrate the theory with examples on the projective plane, the quadric surface and some general type surfaces. This is joint work with Jack Huizenga.

Zhiwei Yun (Stanford)
April 29, 2016

Intersection numbers and higher derivatives of L-functions for function fields

In joint work with Wei Zhang, we prove a higher derivative analogue of the Waldspurger formula and the Gross-Zagier formula in the function field setting under some unramifiedness assumptions. Our formula relates the self-intersection number of certain cycles on the moduli of Drinfeld Shtukas for GL(2) to higher derivatives of automorphic L-functions for GL(2).

Shouwu Zhang (Princeton)
Thursday, May 12, 2016, 4:30 pm, 380-W

Lecture 1 of Distinguished Lecture Series: Rational points: ABC conjecture and Szpiro’s conjecture

I will talk about ABC conjecture and its relation to Szpiro’s conjecture, Milnor’s proof of Szpiro’s inequality (before it was known as Szpiro’s conjecture) and its relation to hyperbolic geometry, and relation between Szpiro’s conjecture and Landau—Siegel conjecture on zeros of L-functions.

Shouwu Zhang (Princeton)
Friday, May 13, 2016, 2:30 pm, 383-N

Lecture 2 of Distinguished Lecture Series: Torsion points and preperiodic points: Manin—Mumford’s conjecture and its dynamical analogue

I will talk about techniques used in different proofs of Manin-Mumford’s conjecture and its analogue in dynamical system: p-adic rigid geometry (Raynaud), o-minimality geometry (Pila—Zannier), Arakelov geometry (Ullmo—Zhang), and perfectoid geometry (Xie).

Shouwu Zhang (Princeton)
Monday, May 16, 2016, 4 pm, 384-H

Lecture 1 of Distinguished Lecture Series: CM points and derivatives of L-dunctions: Andre—Oort’s conjecture and Colmez’ conjecture

I will talk about recent work of Tsimerman about reducing Andre—Oort’s conjecture to an averaged version of Colmez’ conjecture, and some related work on derivatives of L-functions by Zhiwei Yun and Wei Zhang using Drinfeld’s moduli of Shtukas, and by Xinyi Yuan using Shimura curves.

Ronen Mukamel (Rice)
May 20, 2016 (joint with Informal Geometry and Topology Seminar)

Equations for real multiplication in genus two and applications to Teichmuller curves

I will describe methods for computing with genus two curves whose Jacobians admit real multiplication and discuss applications of these methods to Teichmuller curves. This is joint work with Abhinav Kumar.

Colleen Robles (Duke)
May 27, 2016

Degenerations of Hodge structures

Two motivating questions from algebraic geometry are: how can a smooth projective variety degenerate? and what are the “relations” between two such degenerations? One way to gain insight into these questions is to ask the analogous questions of invariants associated with the smooth projective varieties. In this case, the invariant that we have in mind is a polarized Hodge structure. (And indeed detailed analysis of degenerations of polarized Hodge structures can be used to better understand degeneration of smooth projective varieties, and moduli spaces and their compactifications.)

I will explain how the work of Cattani, Kaplan and Schmid allows us to view a polarized limiting mixed Hodge structure (PLMHS) as a degeneration of a polarized Hodge structure. There is a notion of “polarized relation” between PLMHS that encodes information on how varieties may degenerate within a family. I will give a classification of PLMHS and their polarized relations in terms of Hodge diamonds (discrete data associated with a PLMHS), effectively answering the Hodge-theoretic analogs of the two motivating questions above. This is joint work with Matt Kerr.

(The seminar webpage is here.)

Jarod Alper (Australian National University)
September 26, 2014

Associated forms in classical invariant theory

There is an interesting map which assigns to a homogeneous form on of degree with non-vanishing discriminant, a certain form on of degree , which is the Macaulay inverse system of the Milnor algebra of .  It was conjectured in a recent paper by M. Eastwood and A. Isaev that all absolute classical invariants of forms on of degree can be extracted from those of forms of degree via this map.  This surprising conjecture was motivated by the well-known Mather-Yau theorem for isolated hypersurface singularities.  I will report on joint work with A. Isaev which settles this conjecture in full generality and proves a stronger statement in the case of binary forms.

Jim Bryan (University of British Columbia)
October 24, 2014

Donaldson-Thomas theory of local elliptic surfaces via the topological vertex

Donaldson-Thomas (DT) invariants of a Calabi-Yau threefold X are fundamental quantum invariants given by (weighted) Euler characteristics of the Hilbert schemes of X. We compute these invariants for the case where X is a so-called local elliptic surface — it is the total space of the canonical line bundle over an elliptic surface. We find that the generating functions for the invariants admit a nice product structure. We introduce a new technique which allows us to use the topological vertex in this computation — a tool which previously could only be used for toric threefolds. As a by product, we discover surprising new identities for the topological vertex. This is joint work with Martijn Kool, with an assist from Ben Young.

Nasko Atanasov (Harvard)
October 31, 2014

Interpolation and vector bundles on curves

We aim to address the following: When is there a (smooth) curve of degree and genus passing through general points in . Generalizations ask for the dimension of such curves, or replace the point incidence conditions with higher dimensional linear spaces. We will start by relating these statements to a property of the normal bundle of curves in projective space. Next, we will show how to address these questions for and . The demonstrated techniques generalize significantly and lead to an answer to our question for . This is joint work with E. Larson and D. Yang.

Yefeng Shen (Stanford)
November 7, 2014

WDVV equations and Ramanujan identities

The occurrence of modular forms and quasi-modular forms in Gromov-Witten theory is an interesting phenomenon. I will present the following work joint with Jie Zhou. We show that the WDVV equations for elliptic orbifolds are equivalent to the Ramanujan identities for some modular groups. We then apply this to prove the genus zero Gromov-Witten correlation functions for all elliptic orbifolds are quasi-modular forms. Combining with the tautological relations on the moduli space of pointed curves, we also obtain the modularity for all genera. This generalizes an earlier result of Milanov–Ruan and solves a modularity conjecture for the Gromov-Witten theory of the elliptic orbifold curve with four -orbifold points.

Yiwei Shi (University of Chicago)
November 14, 2014

The Shafarevich conjecture for K3 surfaces

Let K be a number field, S a finite set of places of K, and g a positive integer. Shafarevich made the following conjecture for higher genus curves: the set of isomorphism classes of genus g curves defined over K and with good reduction outside of S is finite. In 1983, Faltings proved this conjecture for curves and the analogous conjecture for polarized abelian surfaces. Building on the work of Faltings and Andre, we prove the stronger unpolarized Shafarevich conjecture for K3 surfaces.

Sabin Cautis (University of British Columbia)
November 21, 2014

Categorical Heisenberg actions on Hilbert schemes of points

We define actions of certain Heisenberg algebras on the Hilbert schemes of points on ALE surfaces. This lifts constructions of Nakajima and Grojnowski from cohomology to K-theory and derived categories of coherent sheaves. This action can be used to define Lie algebra actions (using categorical vertex operators) and subsequently braid group actions and knot invariants.

François Charles (Paris Sud and MIT)
December 5, 2014, 1:15 pm (unusual time!)

Geometric boundedness results for K3 surfaces

Tate’s conjecture for divisors on algebraic varieties can be rephrased as a finiteness statement for certain families of polarized varieties with unbounded degrees. In the case of abelian varieties, the geometric part of these finiteness statements is contained in Zarhin’s trick. We will discuss such geometric boundedness statements for K3 surfaces over arbitrary fields and holomorphic symplectic varieties, with application to direct proofs of the Tate conjecture for K3 surfaces that do not involve the Kuga-Satake correspondence.

Dan Edidin (Missouri)
January 9, 2015

Strong regular embeddings and the geometry of hypertoric stacks

We explain how the notion of “strong regular embeddings” can be used to compare the geometry of a stack to that of a regularly embedded substack. This theory can be applied to understand the relationship between singular hypertoric varieties and singular Lawrence toric varieties. While this talk is about stacks, the motivating ideas come from simple observations about invariant rings for actions of finite groups.

Evan O’Dorney (Harvard)
January 16, 2015

Canonical rings of -divisors on

The canonical ring of a divisor on a projective curve is a natural object of study; when is a -divisor, it has connections to projective embeddings of stacky curves and rings of modular forms. I will speak about my results from last summer’s Emory REU concerning the generators and relations of for the simplest curve . When contains at most two points, I give a complete description of ; for general , I give bounds on the generators and relations. I have also proved that the generators (for at most five points) and a Groebner basis of relations between them (for at most four points) depend only on the coefficients in the divisor , not its points or the characteristic of the ground field, and I conjecture that the minimal system of relations varies in a similar way.

Mark Shoemaker (Utah)
January 23, 2015

A proof of the LG/CY correspondence via the crepant resolution conjecture

Given a homogeneous degree five polynomial in the variables , …, , we may view as defining a quintic hypersurface in or alternatively, as defining a singularity in where the group action is diagonal. In the first case, one may consider the Gromov-Witten invariants of . In the second case, there is a way to construct analogous invariants, called FJRW invariants, of the singularity. The LG/CY correspondence states that these two sets of invariants are related. In this talk I will explain this correspondence and its relation to a much older conjecture, the crepant resolution conjecture (CRC). I will sketch a proof that the CRC is equivalent to the LG/CY correspondence in certain cases using a generalization of the “quantum Serre-duality” of Coates-Givental. This work is joint with Y.-P. Lee and Nathan Priddis.

Radu Laza (Stony Brook)
January 30, 2015, 3:20-4:20

Moduli of degree 4 K3 surfaces revisited

For low degree K3 surfaces there are several way of constructing and compactifying the moduli space (via period maps, via GIT, or via KSBA). In the case of degree 2 K3 surface, the relationship between various compactifications is well understood by work of Shah, Looijenga, and others. I will report on work in progress with K. O’Grady which aims to give similar complete description for degree 4 K3s.

Jack Hall (ANU)
January 30, 2015, 3:30-4:30

A generalization of Luna’s etale slice theorem

Let be an affine -variety with an action of a reductive group . A beautiful theorem of Luna is that if has closed -orbit, then there is a -invariant and affine subvariety containing , where is the stabilizer of , such that the induced map is -etale. This result is very useful in applications. Unfortunately, direct extensions of Luna’s result to non-affine, singular varieties are not possible. By slightly weakening the conclusion, however, I will describe a generalization of Luna’s etale slice theorem covering these situations, and many more. This is joint work with Alper (ANU) and Rydh (KTH).

Alessandro Chiodo (Jussieu)
February 13, 2015

Néron models of Picard groups by Picard groups

The Néron model provides a universal extension over a discrete valuation ring R of the degree-zero part Pic⁰C_K of the Picard group of a smooth curve C_K over K=Frac(R). It is natural to exploit the (relative) Picard functor Pic⁰C_R of a regular (semi)stable reduction C_R to describe its Néron model N(Pic⁰C_K). The group Pic⁰C_R is not separated in general, but the Néron model N(Pic⁰C_K) equals Pic⁰C_R modulo the closure of the zero section of Pic⁰C_K (Raynaud, 1970). In some very special cases, we obtain N(Pic⁰C_K) without passing through the quotient by simply singling out the identity component of Pic⁰C_R, that is the group of line bundles of degree 0 on all irreducible components. In general, the quotient does not possess a similar modular interpretation but this talk shows that the Néron model does represent a separated Picard functor of degree-0 line bundles on all irreducible components as soon as we adopt a stack-theoretic stable model of C_K.

Daniel Murfet (USC)
February 20, 2015

Computing with hypersurfaces

Associated to an isolated hypersurface singularity is a triangulated category of matrix factorisations. A decade ago Khovanov and Rozansky figured out how to model knots in a bicategory built out of these triangulated categories. Inspired by their construction I will describe how matrix factorisations can be used to construct a model of the polymorphic lambda calculus of Girard, an abstract functional programming language which underlies, for example, Haskell. Beyond the sheer amusement of representing programs by objects of triangulated categories, there are motivations from within logic and computer science that I will also discuss.
Of course I will explain all the logic/comp-sci terms!

Srinivas (Tata Institute for Fundamental Research)
February 27, 2015, 3:20-4:20

Etale motivic cohomology and algebraic cycles

This talk will report on joint work with A. Rosenschon. There are examples (which I’ll briefly discuss) showing that the torsion and co-torsion of Chow groups are complicated, in general, except in the “classical” cases (divisors and 0-cycles, and torsion in codimension 2); also the integral Hodge conjecture is known to fail. Instead, we may (following Lichtenbaum)
consider the etale Chow groups, which coincide with the usual ones if we use rational coefficients; we show that they have better “integral” properties if we work over the complex numbers. In contrast, they can have infinite torsion in some arithmetic situations (the usual Chow groups are conjectured to be finitely generated).

Sándor Kovács (Washington)
February 27, 2015, 4:30-5:30

Projectivity of the moduli space of stable log-varieties

Stable log-varieties are the higher dimensional analogues of stable pointed curves and their moduli spaces are generalizations of . Just as in the curve case, extending the moduli problem to include stable objects leads to a compactification of the moduli space of the smooth ones. (These are known facts due to work of Koll​ár, Viehweg, Koll​ár-Shepherd-Barron, Alexeev, Keel-Mori, Hacon-McKernan-Xu and others). In the higher dimensional case there are many issues that come up that do not appear in dimension one. Furthermore, the log case is far from a straightforward generalization of the absolute case. In particular, Koll​ár’s powerful Ampleness Lemma does not apply in the log case. This talk is an overview report on recent and ongoing work, joint with Zsolt Patakfalvi, in which we prove a number of results: a generalization of Koll​ár’s Ampleness Lemma that works in the log case, various positivity results on pushforwards of relative pluricanonical sheaves and their determinants and as applications of these we prove that any complete moduli space of stable log-varieties is projective and we establish log-subadditvity of Kodaira dimension of log canonical fiber spaces.

Jie Zhou (Perimeter Institute)
April 3, 2015

Counting higher genus curves in local via mirror symmetry

I will talk about the modularity of the generating series of Gromov-Witten invariants of local by using mirror symmetry.

The moduli space involved in the mirror side of local is naturally identified with a modular curve of genus zero. This implies that the mirror of the generating series, as sections of line bundles over the modular curve, are modular forms. The Fricke involution acting on the modular curve exchanges the asymptotic behaviour at the cusps of the modular forms. It induces a duality of the underling physics theories, in a way similar to the electro-magnetic duality in Seiberg-Witten theory.

The talk is based on a joint work with M. Alim, E. Scheidegger and S.-T. Yau.

Y.P. Lee (Utah)
April 10, 2015

A + B model in conifold transitions for Calabi–Yau threefolds

Witten named two topological field theories for Calabi–Yau threefolds A and B models respectively in his explanation of mirror symmetry. In this talk, A model is the Gromov–Witten theory and B model the variation of Hodge structure. All known examples of (simply-connected) Calabi–Yau threefolds are connected by a special kind of surgery, called (extremal) transition, of which the basic case is the conifold transition.

In this talk, I will explain a phenomenon of partial exchange of A and B models when the Calabi–Yau threefold undergoes a conifold transition. This suggests the possibility of an A+B theory which is invariant under transitions and is therefore equivalent for all (simply-connected) Calabi–Yau threefolds.

This talk is based on joint work with H.-W. Lin and C.-L. Wang.

Anand Patel (Boston College)
April 17, 2015

Algebraic cohomology of Hurwitz spaces

I will introduce the general problem of determining Chow rings of the Hurwitz spaces parametrizing branched covers of the projective line. I will present some conjectures as well as recent progress in the degree 3 case (joint with R. Vakil). This talk may be of interest to topologists as well – I will discuss ongoing work on the stabilization of the algebraic cohomology of Hurwitz spaces as the degree and genus parameters tend to infinity.

Simion Filip (Chicago)
May 1, 2015

Hodge theory and arithmetic in Teichmuller dynamics

The dynamics of a billiard ball in a polygon is a classical dynamical system for which many questions remain open. These questions are closely related to a natural action of the group SL(2,R) on the tangent bundle to the moduli space of Riemann surfaces. It can be viewed as a “complexified” geodesic flow. By recent results of Eskin and Mirzakhani, this action of SL(2,R) enjoys rigidity properties akin to Ratner’s theorems – in particular, orbit closures are submanifolds. In this talk, I will explain why these orbit closures are in fact algebraic varieties with interesting arithmetic properties. For instance, they parametrize algebraic curves with real multiplication and torsion conditions on (factors of) their Jacobian. These results depend on extending results about variations of Hodge structures to this special setting (in particular, giving a different approach to some results of Schmid). No background in any of the above topics will be assumed and I will provide the necessary introduction.

Adrian Zahariuc (Harvard)
Monday May 4, 2015

Specialization of Quintic Threefolds to the Secant Variety

We consider the degeneration of quintic threefolds obtained by running semistable reduction on a general pencil of quintics specializing to the secant variety of a normal elliptic curve in projective 4-space. The question we would like to address is: what are the flat limits of the rational curves in this degeneration? In this talk, I will provide a good “first approximation” to the answer by describing the space of genus zero stable morphisms to the central fiber (as defined by J. Li).

Joe Rabinoff (Georgia Tech)
May 8, 2015

Integration on wide opens and uniform Manin-Mumford

Coleman’s effective version of Chabauty’s method of attacking the Mordell conjecture involves counting zeros of certain p-adic integrals on p-adic open discs, using a Newton polygon argument. Recently, Stoll extended the Chabauty–Coleman method using integration on open discs as well as open annuli, to prove a uniform Mordell conjecture for hyperelliptic curves of fixed genus and small Mordell–Weil rank. We use integration on so-called basic wide open subdomains, the p-adic analogue of a “pair of pants” in a Riemann surface, in order to prove a uniform Manin–Mumford result for curves of a fixed genus with sufficiently degenerate reduction type. This variant of the Chabauty–Coleman method is unique as it can be used to bound *geometric* points on a curve, not just rational points. Other ingredients include potential theory on Berkovich curves, the theory of linear systems on metric graphs, and a comparison of different p-adic integration theories. A special case of this method extends Stoll’s results to all curves of low Mordell–Weil rank.

This work is joint with Eric Katz and David Zureick-Brown. It complements David’s talk on April 13, but it is self-contained and assumes no prior knowledge of Berkovich spaces or the Chabauty–Coleman method.

Andrei Caldararu (Wisconsin)
May 15, 2015

Algebraic proofs of degenerations of Hodge-de Rham complexes

In the first half of the talk I shall present a new algebraic proof of a result of Deligne-Illusie about the degeneration of the Hodge-de Rham spectral sequence. The idea is to reduce the main technical point of their proof to a question about the formality of a derived intersection in an Azumaya space.

In the second half of the talk I shall discuss the main technical difficulty that arises when trying to extend these techniques to obtain a proof of a famous claim of Barannikov-Kontsevich. This claim, which was first proved by analytic methods by Sabbah, expresses the hypercohomology of the twisted de Rham complex in terms of computations with coherent sheaves. It is conceptually the analogue of the Hodge-de Rham degeneration statement for dg categories of matrix factorizations.

This is joint work with Dima Arinkin and Marton Hablicsek.

Alex Perry (Harvard)
May 29, 2015

Derived categories of Gushel–Mukai varieties

A Fano Gushel–Mukai (GM) variety is a Fano variety of Picard number 1, degree 10, and coindex 3. For such a variety , there is a special subcategory of its derived category, which appears to be closely related to its birational geometry. I will discuss a duality operation that exchanges GM varieties of different dimensions and identifies their categories. As a special case, this gives a family of rational GM fourfolds whose category is equivalent to the derived category of a K3 surface. I will also discuss a relation between the derived categories of GM and cubic fourfolds. This is joint work with Alexander Kuznetsov.

Tom Coates (Imperial College London)
June 1, 2015, 4 pm (special algebraic and symplectic geometry seminar)

Mirror Symmetry and Fano Manifolds

Fano manifolds are basic building blocks in algebraic geometry, and the classification of Fano manifolds is a long-standing and important open problem. We explain a surprising connection between Mirror Symmetry and Fano classification. This is joint work with Akhtar, Corti, Galkin, Golyshev, Heuberger, Kasprzyk, Oneto, Petracci, Prince, and Tveiten.

(The seminar webpage is here.)

Alexei Oblomkov (U Mass Amherst)
October 4, 2013

Plane curve singularities, knot homology and Hilbert scheme of points on plane

I will present a conjectural formula for the Poincare polynomial of the Hilbert scheme of points on a planar curve (joint with Rasmussen and Shende).  The formula is written in terms of the Khovanov-Rozansky invariants of the links of the singularities of the curve.  In the case of toric curve the Poincare polynomial also could be written in terms of equivariant Euler characteristic of some sheaf of some particular equivariant sheaf on (joint with Yun). If time permits I will also discuss cohomology ring of the compactified Jacobian of the toric curve and the conjectural description of the ring for general curve singularity (joint with Yun).

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Francois Greer (Stanford)
October 25, 2013

Picard Groups of Moduli Spaces of K3 Surfaces

Polarized K3 surfaces of genus can be thought of as families of canonical curves.  As such, their moduli space has similar properties to .  For instance, both are unirational for low values of g, and both have discrete Picard group.  In this talk, we will use the explicit unirationality of in the range of Mukai models to compute its Picard number, and verify the Noether-Lefschetz conjecture for genus up to 10.  This is joint work with Zhiyuan Li and Zhiyu Tian.

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Bhargav Bhatt (IAS)
November 1, 2013

Lefschetz for local Picard groups

A classical theorem of Lefschetz asserts that non-trivial line bundles on a smooth projective variety of dimension remain non-trivial upon restriction to an ample divisor. In SGA2, Grothendieck recast this result in purely local algebraic terms. Answering a question raised recently by Koll\’ar, we will explain how this local reformulation remains true under milder hypotheses than those imposed in SGA2. Our approach relies on a vanishing theorem in characteristic p, and formal geometry over certain very large (non-noetherian) schemes. This is joint work with Johan de Jong.

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Mark de Cataldo (Stony Brook University)
November 15, 2013

The projectors of the decomposition theorem are absolute Hodge

I report on joint work with Luca Migliorini at Bologna. If you have a map of complex projective manifolds, then the rational cohomology of the domain splits into a direct sum of pieces in a way dictated by the singularities of the map. By Poincaré duality, the corresponding projections can be viewed as cohomology classes (projectors) on the self-product of the domain. These projectors are Hodge classes, i.e. rational and of type (p,p) for the Hodge decomposition. Take the same situation after application of an automorphism of the ground field of complex numbers. The new projectors are of course Hodge classes. On the other hand, you can also transplant, using the field automorphism, the old projectors into the new situation and it is not clear that the new projectors and the transplants of the old projectors coincide. We prove they do, thus proving that the projectors are absolute Hodge classes, i.e. their being of Hodge type survives the totally discontinuous process of a field automorphism.  We also prove that these projectors are motivated in the sense of Andre.

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Allen Knutson (Cornell)
November 22, 2013

Conormal varieties and the Temperley-Lieb algebra

Each permutation pi has an associated determinantal variety called its “matrix Schubert variety”. If one degenerates these determinants to monomials, the components of the resulting scheme are naturally indexed by reduced words for pi as a product of simple reflections ([K-Miller 2005]). It is particularly notable that unlike in most limits of this sort, the resulting scheme has no multiplicities on its components.

I’ll describe the “conormal variety” to a variety, which shows up in many contexts (e.g. projective duality), and extend this degeneration idea to the conormal varieties of the matrix Schubert varieties. The limit scheme now includes the conormal variety to the original limit, and to its projective dual, but also some fundamentally new components which appear with multiplicity. Then I’ll state some conjectures, in particular that the components are naturally indexed by words in the generators of the Temperley-Lieb algebra, from which one can also predict the multiplicities.

This work is joint with Paul Zinn-Justin.

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Yuri Tschinkel (Courant (=NYU) and the Simons Foundation)
Monday, December 2, 2013, 2:30 pm (note unusual date and time)

Igusa integrals

Geometric Igusa integrals appear as important technical tools in the study of rational and integral points on algebraic varieties. I will describe some of these applications (joint work with A. Chambert-Loir).

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Nick Katz (Princeton)
January 31, 2014

Equidistribution questions arising from universal extensions

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Akhil Mathew (Harvard)
Tuesday February 18, 4 pm, 383-N (joint with topology)

The Galois group of a stable homotopy theory

To a “stable homotopy theory” (a presentable, symmetric monoidal stable ∞-category), we naturally associate a category of finite ́etale algebra objects and, using Grothendieck’s categorical machine, a profinite group that we call the Galois group. This construction builds on, and generalizes, ideas of many authors, and includes the ́etale fundamental group of algebraic geometry as a special case. We calculate the Galois groups in several examples, both in settings of rational and p-adic homotopy and in “chromatic” stable homotopy theories. For instance, we show that the Galois group of the periodic E∞-algebra of topological modular forms is trivial, and, extending work of Baker and Richter, that the Galois group of K(n)-local stable homotopy theory is an extended version of the Morava stabilizer group.

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Bernd Sturmfels (Berkeley)
February 28, 2014

The Euclidean Distance Degree of an Algebraic Variety

The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. The Euclidean distance degree is the number of critical points of this optimization problem. We focus on varieties encountered in engineering applications, and we discuss exact computational methods. Our running example is the Eckart-Young Theorem which states that the nearest point map for low rank matrices is given by the singular value decomposition. This is joint work with Jan Draisma, Emil Horobet, Giorgio Ottaviani, Rekha Thomas.

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Kiran Kedlaya (UCSD)
March 7, 2014

Sato-Tate groups of abelian surfaces

There are a number of theorems and conjectures in arithmetic geometry which control the structure of an abelian variety over a number field in terms of its reductions to various finite fields. Here we consider a rather crude question along these lines: how much structure can be read off by computing the zeta functions of these reductions and retaining only the statistical behavior of these zeta functions as one averages over primes? For elliptic curves, this is known (under appropriate conjectures, which are known in many cases) to distinguish whether or not the curve has complex multiplication, and if so whether this happens over the base field or an extension field. We describe the corresponding picture for genus 2 curves: the punchline is that there are 52 different possible cases, corresponding to Galois module structures on real endomorphism algebras. These cases will be illustrated with some pretty pictures! Joint work with Francesc Fite, Victor Rotger, and Andrew Sutherland.

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June Huh (Michigan)
May 2, 2014

Homology classes in algebraic varieties: nef, effective, and prime

The homology group of an algebraic variety is an abelian group equipped with several additional structures. It contains the set of primes, the homology classes of subvarieties. It contains the semigroup of effective classes, the nonnegative linear combinations of primes. It contains the semigroup of nef classes, the classes which intersect all primes of complementary dimension nonnegatively. We will see how these subsets look like in a particular algebraic variety, the one associated to the polytope `permutahedron’. The semigroups in this case have tractable structures, while the distribution of primes is more mysterious and related to some deep combinatorial conjectures on matroids.

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Zhiyuan Li (Stanford)
May 9, 2014

Special cycles on Shimura varieties of orthogonal type

Let be a connected smooth Shimura variety. There are many special algebraic cycles coming from sub-Shimura variety of the same type in all codimensions, called special cycles on . A Hodge type question is whether these special cycles exhaust all the low degree cohomology classes of . In this talk, I will discuss relation between this question and Arthur’s theory and briefly talk about the work of Beregon-Millson-Moeglin in this direction.

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Qile Chen (Columbia)
May 16, 2014

-curves on quasi-projective varieties

The theory of stable log maps was developed recently for studying the degeneration of Gromov-Witten invariants. In this talk, I will introduce another interesting aspect of stable log maps as a useful tool for investigating -curves on quasi-projective varieties, which are the analogue of rational curves on proper varieties. At least two applications of -curves will be discussed in this talk. For classical birational geometry, the -curves can be used to produce very free rational curves on general Fano complete intersections in projective spaces. On the arithmetic side, -connectedness gives a general frame work for the existence of integral points over function field of curves. This is a joint work in progress with Yi Zhu.

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Aaron Pixton (Clay)
May 23, 2014

Double ramification cycles and tautological relations

Double ramification cycles parametrize curves that admit maps to the projective line with specified ramification over zero and infinity. They can be extended to the moduli space of stable curves by using the virtual class in relative Gromov-Witten theory. I will describe a conjectural formula for these extensions in terms of tautological classes. The formula is motivated by a connection with recent joint work with Pandharipande and Zvonkine on Witten’s r-spin class, and it comes with a family of tautological relations which extend relations studied by Grushevsky and Zakharov.

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John Lesieutre (MIT)
May 30, 2014

Negative answers to some positivity questions

I will explain how the classically-studied action of Cremona transformations on configurations of points in projective space can be used to construct several counterexamples in birational geometry: nefness is not an open condition in families, Zariski decompositions do not always exist in dimension 3, and a variety can have infinitely many Fourier-Mukai partners. As time permits, I’ll discuss some related examples on Calabi-Yau threefolds.

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Fridays 3:15-4:15 in 383-N (with exceptions)

Click on the title to see the abstract (if available).  (For earlier talks in this seminar, click here. For related seminars, click here.) For more information, please contact Jun Li, Zhiyuan Li, or Ravi Vakil.

(The seminar webpage is here.)

Zhiyuan Li (Stanford)
September 24, 2012

Modular forms and special cubic fourfolds

A special cubic fourfold is a smooth cubic hypersurface in containing a rational normal scroll. Brendan Hassett has shown that the sets of special cubic fourfolds form divisors in the parameterization space . A classical geometry question is to know the degrees of these divisors in .  We solve this problem by showing that the generating series of the degree of these divisors is a modular form with respect to a level 3 congruence subgroup of . The modularity comes from the work of Borcherds’ generalized Gross-Kohnen-Zagier theorem. This is joint work with Letao Zhang.

Hsian-Hua Tseng (Ohio State)
October 1, 2012

Toric mirror maps revisited

For a compact semi-Fano toric manifold , the toric mirror theorem of Givental and Lian-Liu-Yau says that a generating function of 1-point genus 0 descendant Gromov-Witten invariants, the -function of , coincides up to a mirror map with a function which is written using the combinatorics of . The procedure of obtaining the mirror map, which involves expanding as a suitable power series, is somewhat mysterious. In this talk we’ll describe some attempts at understanding the mirror maps more geometrically.

Daniel Erman (Michigan)
October 8, 2012

Semiample Bertini Theorems over finite fields

We prove a semiample generalization of Poonen’s Bertini Theorem over a finite
field that implies the existence of smooth sections for wide new classes of divisors. The probability of smoothness is computed as a product of local probabilities taken over the fibers of a corresponding morphism. This joint with Melanie Matchett Wood.

Sam Grushevsky (Stony Brook)
October 15, 2012

The stable cohomology of moduli of abelian varieties and compactifications

The stable (for much large than the degree) cohomology of the moduli space of principally polarized abelian varieties was computed by Borel, as the group cohomology of the symplectic group. Using topological methods, Charney and Lee computed the cohomology of the Satake-Baily-Borel compactification of . In this talk we will discuss the question of computing the stable cohomology of toroidal compactifications, and in particular will discuss the stabilization of suitable cohomology for the perfect cone toroidal compactification, and explicit computation of some of these stable cohomology groups, by algebraic methods. Joint work in progress with Klaus Hulek and Orsola Tommasi (there is also an independent related work in progress by Jeff Giansiracusa and Gregory Sankaran, by topological methods).

Sam Payne (Yale)
Friday, October 26, 2012, 2:30 pm, 383-N

Tropicalization of the moduli space of curves

The analogy between moduli spaces of stable curves and more recently constructed moduli spaces of tropical curves is explained, canonically and functorially, through a generalized tropicalization map for toroidal embeddings of Deligne-Mumford stacks. This is joint work with D. Abramovich and L. Caporaso.

Jordan Ellenberg (Wisconsin)
October 29, 2012

Geometric analytic number theory

Analytic number theory has traditionally focused on questions about distribution of various arithmetic objects (prime numbers, ideal class groups, rational points on varieties) over . Many of the standard questions in that field can be expressed just as easily over a global field of characteristic , such as . It turns out that analytic number theory in this setting often reveals unexpected connections with algebraic geometry and algebraic topology, with information flowing in both directions. I will survey some recent progress in this area, focusing on the work of myself, Akshay Venkatesh, and Craig Westerland on the Cohen-Lenstra conjectures over function fields and its relation with the stable cohomology of Hurwitz spaces. Time permitting, I will also say something about the Hardy-Littlewood method and its relationship with moduli spaces of rational curves on varieties (work of Pugin and Lee) and speculate a little about the connection between this story and “motivic analytic number theory” (a la Ravi Vakil, Melanie Matchett Wood, Dan Erman, and Daniel Litt, among others) and the theory of -modules (me, Tom Church, and Benson Farb, among others.) Time will not actually permit all this, so let this abstract serve as a list of things I would be interested in talking with people about during my visit, whether or not they appear during my seminar.

Bhargav Bhatt (Michigan/IAS)
Friday, November 9, 2012, 2:30 pm, 383-N

p-adic derived de Rham cohomology

A basic theorem in Hodge theory is the isomorphism between de Rham and Betti cohomology for complex manifolds; this follows directly from the Poincare lemma. The p-adic analogue of this comparison lies deeper, and was the subject of a series of extremely influential conjectures made by Fontaine in the early 80s (which have since been established by various mathematicians). In my talk, I will first discuss the geometric motivation behind Fontaine’s conjectures, and then explain a simple new proof based on general principles in derived algebraic geometry — specifically, derived de Rham cohomology — and some classical geometry with curve fibrations. This work builds on ideas of Beilinson who proved the de Rham comparison conjecture this way.

John Ottem (Cambridge)
November 12

Ample subschemes and partially positive line bundles

We introduce a notion of ampleness for subschemes of higher codimension and investigate geometric properties satisfied by ample subschemes, e.g. the Lefschetz hyperplane theorems and numerical positivity. Using these results, we also construct a counterexample to a conjecture of Demailly-Peternell-Schneider.

Yunfeng Jiang (Imperial College London)
November 26

On the crepant transformation conjecture

Let and be two smooth Deligne-Mumford stacks. We call dash arrow a crepant transformation if there exists a third smooth Deligne-Mumford stack and two morphisms , such that the pullbacks of canonical divisors are equivalent, i.e. . The crepant transformation conjecture says that the Gromov-Witten theory of and is equivalent if is a crepant transformation. This conjecture was well studied in two cases: the first one is the case when and are both smooth varieties; the other is the case that there is a real morphism to the coarse moduli space of , resolving the singularities of . In this talk I will present some recent progress for this conjecture, especially in the case when both and are smooth Deligne-Mumford stacks.

Matt Satriano (University of Michigan)
December 3 (in 380-Y)

Toric Stacks and Applications to Cycle Theory

No prior knowledge of stacks will be assumed for this talk. We will discuss the theory of toric stacks, which like toric varieties, is a class of stacks which is particularly amenable to computation. Using the theory developed, we will show that it is impossible to construct a degree map for 0-cycles on even the nicest of Artin stacks. This is based on joint work with Dan Edidin and Anton Geraschenko.

János Kollár (Princeton)
Thursday January 10, 4:15-5:15, 380-W

MRC Distinguished Lecture 1 of 2: Local topology of analytic spaces

Let be a subset of defined as the common zero set of some holomorphic functions. What can one say about the local structure of ? Each point has an open neighborhood that is a cone over an (odd real dimensional) topological space called the link of . If is a smooth point of complex dimension , the link is a sphere of real dimension .

Our main interest is to understand the most complicated links. The general answer is not known but we show how to construct examples where the fundamental group of a link is nearly arbitrary. Joint work with Misha Kapovich.

János Kollár (Princeton)
Thursday January 24, 3:15-4:15, Gates B12

MRC Distinguished Lecture 2 of 2: How to recognize families of Cartier divisors?

We propose a strengthening of the Grothendieck-Lefschetz hyperplane theorem for the local Picard group, prove some special cases and derive several consequences to the deformation theory of log canonical singularities.

Amnon Yekutieli (Ben-Gurion Univ.)
February 4

Residues and duality for schemes and stacks

Let K be a regular noetherian commutative ring. I will begin by explaining the theory of rigid residue complexes over essentially finite type K-algebras, that was developed by J. Zhang and myself several years ago. Then I will talk about the geometrization of this theory: rigid residue complexes over finite type K-schemes. An important feature is that the rigid residue complex over a scheme X is a quasi-coherent sheaf in the etale topology of X. For any map between K-schemes there is a rigid trace homomorphism (that usually does not commute with the differentials). When the map of schemes is proper, the rigid trace does commute with the differentials (this is the Residue Theorem), and it induces Grothendieck Duality.

Then I will move to finite type Deligne-Mumford K-stacks. Any such stack has a rigid residue complex on it, and for any map between stacks there is a trace homomorphism. These facts are rather easy consequences of the corresponding facts for schemes, together with etale descent. I will finish by presenting two conjectures, that refer to Grothendieck duality for proper maps between DM stacks. A key condition here is that of tame map of stacks.

Xuanyu Pan (Columbia)
February 4

The Geometry of Moduli space of rational Curves on Complete intersections

For a low degree smooth complete intersection X, we consider the general fiber F of the following evaluation map ev of Kontsevich moduli space. I will give general structure theorems of F. It answers questions relating to
(1) Rational connectedness of moduli space
(2) Enumerative geometry
(3) The Search for a new 2-Fano variety
(4) Picard group of moduli space

Brendan Hassett (Rice)
February 22 (4 pm, 383-N)

K3 surfaces, level structure, and rational points

Moduli spaces of elliptic curves with level structure are fundamental for arithmetic and Diophantine problems over number fields in particular. For K3 surfaces, the Brauer group plays the role of the torsion points. Recently, a number of papers have systematically investigated how the Brauer group may be used to formulate criteria for the existence of rational points. However, geometric interpretations for the Brauer elements remain elusive. We present examples of several such interpretations and arithmetic applications, with a view toward putting these in a larger framework.

(joint work with V’arilly-Alvarado)

Chenyang Xu (Beijing)
February 25

Comparison of stabilities

There are different stability notions coming from different ideas of constructing the moduli space. In this talk, I will try to explore the relations among KSBA-stability, K-stabilty and asymptotic GIT stability. Smooth canonical polarized varieties are stable in all these senses, however the limiting behaviors turn out to be different. As a corollary, we answer the longstanding question that asymptotically GIT Chow semistable varieties does not form a proper family. (joint with Xiaowei Wang)

Runpu Zong (Princeton)
Thursday February 28, 3:15-4:15, Gates B12

Weak Approximation for Isotrivial Family

Following the celebrated Graber-Harris-Starr theorem that any geometrically smooth rationally connected variety over function field of curve has a rational point, there is Hassett-Tschinkel’s conjecture that whether rational points have weak approximation property of these families. While wildly open, we prove this conjecture when the general fibers of the family are isomorphic to each other, which is a large class of families that former results never covered.

Igor Dolgachev (University of Michigan)
March 11

Rational self-maps of moduli varieties

I will discuss some known examples of rational self-maps of finite degree greater than one of moduli varieties considered as stacks defining some moduli problem. A classical example of this sort is a covariant of order d on the space homogeneous forms of degree d. Other examples include moduli varieties of algebraic curves of low genus or abelian varieties with some level structure.

Nathan Ilten (Berkeley)
April 1, 2:45-3:45 pm

Equivariant Vector Bundles on T-Varieties

Klyachko has shown that there is an equivalence of categories between equivariant vector bundles on a toric variety X and collections of filtered vector spaces satisfying some compatibility conditions. I will discuss joint work with H. Suess which generalizes this equivalence to the setting of T-equivariant vector bundles on a normal variety X endowed with an effective action of an algebraic torus T. Indeed, T-equivariant vector bundles on X correspond to collections of filtered vector bundles on a suitable quotient of X.

This correspondence can be applied to show that T-equivariant bundles of low rank on projective space split, as well as to easily compute all global vector fields on rational complexity-one T-varieties.

Anand Deopurkar (Columbia)
April 1, 4-5 pm

Compactifying spaces of branched covers

Moduli spaces of geometrically interesting objects are often non-compact. They need to be compactified by adding some degenerate objects. In many cases, this can be done in several ways, leading to a menagerie of birational models, which are related to each other in interesting ways. In this talk, I will explore this idea for the spaces of branched covers of curves, known as the Hurwitz spaces. I will construct a number of compactifications of these spaces by allowing more and more branch points to coincide. I will describe the geometry of the resulting spaces for the case of triple covers.

Greg G. Smith (Queen’s University)
April 8

Nonnegative sections and sums of squares

A polynomial with real coefficients is nonnegative if it takes on only nonnegative values. For example, any sum of squares is obviously nonnegative. For a homogeneous polynomial with respect to the standard grading, Hilbert famously characterized when every nonnegative homogeneous polynomial is a sum of squares. In this talk, we will examine this converse for line bundles on a totally-real projective subvariety. We will uncover some unexpected connections with classical algebraic geometry and, by working with multigraded polynomial rings, provide many new examples in which every nonnegative homogeneous polynomial is a sum of squares.

Donu Arapura (Purdue University)
April 15, 3:30-4:20 pm, 383-N

Splittable objects in derived categories and Kodaira vanishing

I first want to explain some generalizations of the Kodaira vanishing theorems due to Kollár, Saito and others, which involves some fairly serious Hodge theory. Then I would outline some ideas for approaching these using characteristic p techniques, which is more elementary. Some of this is work in progress.

Vivek Shende (MIT)
April 15, 4:30-5:20 pm, 383-N

Higher Discriminants

Given a map between smooth varieties, the discriminant is the locus of points in such that the fibre is singular. We suggest the following generalization: the th discriminant of is the locus of points such that for any dimensional disc passing through , the fibre is singular at some point of . The sense of this definition is clarified by the following two theorems in the case when is proper and equidimensional: (1) the support of any summand of the (derived) pushforward of the constant sheaf on is equal to a component of a higher discriminant, and (2) the pushforward of the constant function on is equal to an integer linear combination of the Euler obstructions of components of higher discriminants.

Paolo Aluffi (Florida State University)
April 22, 4-5 pm, 381-U

Segre classes of monomial subschemes

Segre classes are fundamental intersection-theoretic invariants: many problems in enumerative geometry may be reduced to computations of Segre classes, and basic invariants such as Milnor numbers and classes may be expressed in terms of Segre classes. We propose a formula computing the Segre class of an arbitrary monomial subscheme, and prove this formula in several representative cases. The formula is reminiscent of intersection computations in terms of mixed volumes of polytopes and convex bodies as in the classical Bernstein’s theorem and more recent work of Kaveh-Khovanskii, Huh, and others.

Zhiyu Tian (Caltech)
April 29, 4-5 pm, 381-U

Weak approximation for cubic hypersurfaces

Given an algebraic variety X over a field F (e.g. number fields, function fields), a natural question is whether the set of rational points X(F) is non-empty. And if it is non-empty, how many rational points are there? In particular, are they Zariski dense? Do they satisfy weak approximation? For cubic hypersurfaces defined over the function field of a complex curve, we know the existence of rational points by Tsen’ s theorem or the Graber-Harris-Starr theorem. In this talk, I will discuss the weak approximation property of such hypersurfaces.

Andrew Morrison (ETH)
May 6, 3:30-4:30pm, 381-U

Behrend’s function is constant on

We will prove that Behrend’s function is constant on . A calculation of motivic zeta functions shows the relevant Milnor fibers have zero Euler characteristic. As a corollary we see that is generically reduced and all its components have the same dimension mod 2. Time permitting we will also extend these results to moduli schemes of curves and points in threefolds coming from resolutions of ADE singularities.

Arnav Tripathy (Stanford)
May 6, 4:45-5:45 pm, 381-U

Stabilisation of symmetric powers

Starting with a scheme , we may consider the sequence of its symmetric powers . I’ll remind the audience of reasons from topology and arithmetic geometry for why we should expect various sorts of asymptotic behaviour of this sequence. I’ll then discuss a certain natural sort of stabilisation, namely that of the etale homotopy groups of this sequence.

Frank Sottile (TAMU)
May 13, 4-5 pm, 381-U

Galois groups of Schubert problems

Work of Jordan from 1870 showed how Galois theory can be applied to enumerative geometry. Hermite earlier showed that the equivalence of Galois groups with geometric monodromy groups, and in 1979 Harris used this to study Galois groups of many enumerative problems. Vakil gave a geometric-combinatorial criterion that implies a Galois group contains the alternating group. With Brooks and Martin del Campo, we used Vakil’s criterion to show that all Schubert problems involving lines have at least alternating Galois group. White and I have given a new proof of this based on 2-transitivity.

My talk will describe this background and sketch a current project to systematically determine Galois groups of all Schubert problems of moderate size on all small classical flag manifolds, investigating at least several million problems. This will use supercomputers employing several overlapping methods, including combinatorial criteria, symbolic computation, and numerical homotopy continuation, and require the development of new algorithms and software.

The algebraic geometry notes used over the last few years are available here.

The 2017-18 talks in the Stanford algebraic geometry are here. Here are the webpages of previous years.

• 2016-17
• 2015-16
• 2014-15
• 2013-14
• 2012-13
• 2011-12
• 2010-11
• 2009-10
• 2008-09
• 2007-08
• 2006-07
• 2005-06
• 2004-05
• 2003-04
• 2002-03
• fall 2001