# Seminars & Events for Algebraic Geometry Seminar

##### F-Signature and Relative Hilbert-Kunz Multiplicity Revisited

In this talk, building on an observation of Yao, we will sketch a partial answer to a question of Watanabe and Yoshida by showing that the F-signature and relative Hilbert-Kunz multiplicity (for cyclic modules) coincide. The method of proof also suggests a number of generalizations of F-signature which we will present as time allows.

##### Hassett-Keel Program in Genus Four

The Hassett-Keel program aims to give modular interpretations of log canonical models for the moduli spaces of curves. The program, while relatively new, has attracted the attention of a number of researchers, and has rapidly become one of the most active areas of research concerning the moduli of curves. In the genus four case, we construct several of these models as GIT quotients of a single, elementary space.

##### Functoriality in Gromov--Witten theory

I will discuss the functoriality problem of Gromov-Witten theory, including the past works on crepant transformation conjecture for ordinary flops. The tools used include standard techniques like localization, as well as novel ones (in GWT) like algebraic cobordism. If time allows, I will mention possible functoriality under extremal transitions. This is a joint project with H.-W. Lin and C.-L. Wang and F. Qu.

##### Derived categories of coherent sheaves on K3 surfaces

I will survey what is known and what is not known about the derived category of coherent sheaves on K3 surfaces. In particular, I shall explain how to rephrase a conjecture of Bridgeland describing the group of autoequivalences in terms of more conventional moduli spaces and period domains.

##### A generic Nakano vanishing theorem

The genenic vanishing theorem of Green-Lazarsfeld says, roughly speaking, that the cohomology of a generic topologically trivial line bundle on a compact Kaehler manifold vanishes below a certain degree that depends only on the Albanese mapping. I will explain how to generalize this to a Nakano-type vanishing theorem and, in the process, talk about a special class of coherent sheaves on abelian varieties.

##### Behavior of Welschinger invariants under Morse simplification

Welschinger invariants, real analogs of genus 0 Gromov-Witten invariants, provide non-trivial lower bounds in real algebraic geometry. In this talk I will explain how to get some wall-crossing formulas relating Welschinger invariants of the same (up to deformation) rational algebraic surface with different real structures. This relation is obtained via a real version of a formula by Abramovich and Bertram which computes Gromov-Witten invariants using deformations of complex structures. It can also be seen as a real version, in our special case, of Ionel and Parker's symplectic sum formula. If time permits, I will give some qualitative consequences of this study, for example the vanishing of Welschinger invariants in some cases, and will discuss some generalizations. This is joint work with Nicolas Puignau (UFRJ, Rio de Janeiro)

##### On the numerical dimension of pseudo-effective divisors in positive characteristic

Suppose that X is a smooth algebraic variety over an algebraically closed field and that D is a pseudo effective R-divisor on X. In characteristic zero, by utilizing vanishing theorems, Nakayama proved that if D is not numerically equivalent to the negative part of its Zariski decomposition, then D is pseudo-effective. We prove the same result in characteristic p > 0 using the Frobenius morphism as a replacement for vanishing theorems. This is joint work with Paolo Cascini, Christopher Hacon and Mircea Mustata.

##### MMP for moduli spaces of sheaves on K3 surfaces and Cone Conjectures

We report on joint work in progress with A. Bayer on how one can use wall-crossing techniques to study the birational geometry of a moduli space M of Gieseker-stable sheaves on a K3 surface X. In particular:

(--) We will give a "modular interpretation" for all minimal models of M.

(--) We will describe the nef cone, the movable cone, and the effective cone of M in terms of the algebraic Mukai lattice of X.

(--) We will establish the so called Tyurin/Bogomolov/Hassett-Tschinkel/Huybrechts/Sawon Conjecture on the existence of Lagrangian fibrations on M.

##### Rationality in families of threefolds

In a joint work with Tommaso de Fernex, we prove that in a family of projective threefolds deﬁned over an algebraically closed ﬁeld, the locus of rational ﬁbers is a countable union of closed subsets of the locus of separably rationally connected ﬁbers. When the ground ﬁeld has characteristic zero, this implies that the locus of rational ﬁbers in a smooth family of projective threefolds is a countable union of closed subsets of the parameter space. General expectation suggests that the result maybe false in higher dimension.

##### Enumeration of singular curves with tangency conditions

How many nodal degree d plane curves are tangent to a given line? The celebrated Caporaso-Harris recursion formula gives a complete answer for any number of nodes, degrees, and all possible tangency conditions. In this talk, I will report my recent work on the generalization of the above problem to count singular curves with given tangency condition to a fixed smooth divisor on general surfaces. I will relate the enumeration to tautological integrals on Hilbert schemes of points and show the numbers of curves in question are given by universal polynomials. As a result, we can obtain infinitely many new formulas for nodal curves and understand the asymptotic behavior for all singular curves with any tangency conditions.

##### A mirror theorem for the mirror quintic

The celebrated Mirror Theorem of Givental and Lian-Liu-Yau states that the A model (quantum cohomology, rational curve counting) of the Fermat quintic threefold is equivalent to the B model (complex deformations, period integrals) of its mirror dual, the mirror quintic orbifold. In order for mirror symmetry to be a true duality however, one must also show that the B model of the Fermat quintic is equivalent to the A model of the mirror quintic. We prove such an equivalence by relating the orbifold Gromov-Witten theory of the mirror quintic to period integrals over a one parameter deformation of the Fermat quintic. This involves new calculations in orbifold Gromov-Witten theory.

##### The holomorphic height pairing

In joint work with Mirel Caibar we show that the Beilinson-Bloch, Gillet-Soulé height pairing between algebraic (n-1)-cycles on a (2n-1)-dimensional complex projective manifold X is the imaginary part of a natural (multivalued) complex quantity that varies holomorphically on components of the Hilbert scheme of X. This pairing is intimately related to the Abel-Jacobi image of the respective cycles. Furthermore this 'holomorphic height pairing' can be extended to a C-infinity pairing on integral currents whose supports are real (2n-2)-dimensional oriented submanifolds of X. As an application in the holomorphic situation, Abel's theorem for Riemann surfaces, suitably interpreted, says that, for zero-cycles algebraically equivalent to zero, Abel-Jacobi equivalence is the same as incidence equivalence.

##### Derived categories and variation of GIT quotients

Semi-orthogonal decompositions of derived categories of coherent sheaves on varieties provide an efficient and flexible means of capturing deep geometric ties to between varieties. I will describe semi-orthogonal decompositions relating the derived categories of GIT quotients obtained via different linearizations of the action. As an application, I will show how to construct full exceptional collections on some moduli spaces of pointed rational curves. This is joint work with D. Favero (Wien) and L. Katzarkov (Miami/Wien). | ||

##### Regular del Pezzo surfaces with irregularity

Over perfect fields, the geometry of regular del Pezzo surfaces has been classified, but over imperfect fields, the problem remains largely open. We construct the first examples of regular del Pezzo surfaces X that have positive irregularity h^1(X, O_X ) > 0. Our construction is by quotienting a regular, quasi-linear surface (i.e. a regular variety that is geometrically a non-reduced first-order neighborhood of a plane) by explicit rank 1 foliations. We also find a restriction on the integer pairs that are possible as the anti-canonical degree and irregularity of such surfaces.

##### Motivic Galois groups and periods

We explain how to associate a universal pro-algebraic group to the Betti realization functor from the triangulated category of motives over a subfield of $\mathbb{C}$. We then give a concrete description of the torsor of isomorphisms between the Betti realization and the de Rham realization. If time permits, some applications to periods will be sketched.

##### A divisor with non-closed diminished base locus

I will explain the construction of a pseudoeffective **R**-divisor *Dλ* on the blow-up of * P3* at nine very general points which has negative intersections with an infinite set of curves, whose union is Zariski dense. It follows that the diminished base locus

*is not closed and that Dλ does not admit a Zariski decomposition in even a very weak sense. Along the way I will discuss some related examples, including an*

**B**-(Dλ) = ∪A ample**B**(Dλ+A)**R**-divisor which is nef on very general fibers of a family, but fails to be nef over countably many prime divisors in the base.

##### Algebraic Geometry of K-stability and its application to Moduli varieties

K-stability is a stability for varieties (Tian, Donaldson), a modification of classical stability (Mumford). While it has been known for several decades that classically stability does not work in higher dimensions moduli construction, the author explains how the K-stability fits into recent construction of compact moduli of general type varieties by KSBA (Koll\'ar-Shepherd-Barron, Alexeev) theory. CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

(PLEASE NOTE SPECIAL DAY AND TIME.)

##### Big cycles and volume functions

The volume of a divisor is an important invariant measuring the "positivity" of its numerical class. I will discuss an analogous construction for cycles of arbitrary codimension. In particular, this yields geometric characterizations of big cycle classes modeled on the well-known criteria for divisor and curve classes.

##### Which morphisms between tropical curves come from algebraic geometry?

A finite morphism between stable marked curves over a non-archimedean field gives rise in a natural way to a morphism of metric graphs (also known as "abstract tropical curves"). PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Birational geometry of $\bar{M}_{0,n}$ and conformal blocks

In the last several decades, conformal blocks have been studied by many algebraic geometers who are interested in the geometry of moduli spaces of vector bundles. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.