Title: Pathologies for Fano varieties and singularities in positive characteristic
Abstract: After the seminal work of Hacon and Xu, a large part of the MMP has been established for threefolds over fields of characteristic p > 5. However very little is known in small characteristic, especially in the study of threefold singularities.
In this talk I will review the cone construction for Fano-type varieties and I will explain how this allows to construct pathological examples of singularities in positive characteristic starting from Fano varieties violating vanishing theorems . As an application, I will show how to construct log del Pezzo surfaces in characteristic two and three violating Kodaira vanishing, thus deducing the existence of klt not Cohen-Macaulay threefold singularities and I will use some recent examples of Totaro to construct non-normal plt centres.
Title: Threefold flops and the derived contraction algebra
Abstract: Derived categories have a strong link to birational geometry -- for example, Bridgeland's famous result that threefold flops induce derived equivalences. Wemyss's Homological Minimal Model Programme is an attempt to run the MMP using derived methods. In particular, given a threefold flopping contraction, one can associate a certain finite-dimensional algebra, the contraction algebra. It controls the noncommutative deformation theory of the flopping curves, and is conjectured to determine completely the geometry of the base. In this talk, I'll describe a natural generalisation, the derived contraction algebra, which controls derived deformations.
Title: Moduli spaces of stable pairs on the resolved conifold
Abstract: Moduli space of Pandharipande-Thomas (PT) stable pairs provide an example of moduli spaces of sheaves whose scheme-theoretic geometry can be described to a reasonable level of details. I would like to present a new example of such moduli scheme.
I would start by a general introduction on enumerative theories, Donaldson-Thomas and PT-invariants, and the relationship between the two. This would be followed by a description of Ferrand's construction to produce, in a controlled way, a double scheme starting from a reduced scheme: this construction is crucial in analysing the PT moduli schemes we are dealing with.
Therefore, I would consider the examples of PT moduli schemes already present in literature, all based on a non-compact Calabi-Yau threefold called "resolved conifold". Finally, I would present my own contribution and show the interesting geometries this new moduli scheme features.Moduli space of Pandharipande-Thomas (PT) stable pairs provide an example of moduli spaces of sheaves whose scheme-theoretic geometry can be described to a reasonable level of details. I would like to present a new example of such moduli scheme.
I would start by a general introduction on enumerative theories, Donaldson-Thomas and PT-invariants, and the relationship between the two. This would be followed by a description of Ferrand's construction to produce, in a controlled way, a double scheme starting from a reduced scheme: this construction is crucial in analysing the PT moduli schemes we are dealing with.
Therefore, I would consider the examples of PT moduli schemes already present in literature, all based on a non-compact Calabi-Yau threefold called "resolved conifold". Finally, I would present my own contribution and show the interesting geometries this new moduli scheme features.
Title: Integral Chow ring of moduli of hyperelliptic curves
Abstract: There is by now an extensive theory of rational Chow rings of moduli spaces of smooth curves. Equivariant intersection theory, as developed by Totaro, Edidin and Graham, can be used to study the integral version of these Chow rings, which are not so well understood. In the first part of the talk I will recall the definition of equivariant Chow ring. In the second part I will focus on the computation, in terms of generators and relations, of the integral Chow ring of the moduli stack of smooth hyperelliptic curves (of odd genus) . Time permitting, I will explain how the same techniques can be used to compute the integral Picard group of moduli stacks of smooth curves of low genus in positive characteristic.
More precisely, I would first introduce the main actors, namely the moduli stack of smooth hyperelliptic curves of odd genus H_g and a scheme X with some interesting properties, and I would show that H_g=[X/GL_3 x C*]. This is done by constructing a "big" family of hyperelliptic curves over X with a "good" action of GL_3 x C*. After that, I would present the final result, i.e. the presentation of the Chow ring of H_g with generators and relations, and I would explain the geometrical meaning of the generators as Chern classes of some vector bundles over H_g.
Time permitting, I would like to talk also about similar results for M_3, the moduli stack of smooth curves of genus 3.In the talk I would like to present the computation of the integral Chow ring of the moduli stack of smooth hyperelliptic curves (of odd genus) that I recently completed, and what is in general the strategy for doing this type of computations.
More precisely, I would first introduce the main actors, namely the moduli stack of smooth hyperelliptic curves of odd genus H_g and a scheme X with some interesting properties, and I would show that H_g=[X/GL_3 x C*]. This is done by constructing a "big" family of hyperelliptic curves over X with a "good" action of GL_3 x C*. After that, I would present the final result, i.e. the presentation of the Chow ring of H_g with generators and relations, and I would explain the geometrical meaning of the generators as Chern classes of some vector bundles over H_g.
Time permitting, I would like to talk also about similar results for M_3, the moduli stack of smooth curves of genus 3.
Title: The Monodromy of a Semistable Family of Curves
Abstract: For a semistable family of curves over the complex numbers, we describe explicitly the monodromy on the cohomology of the generic fiber. Using logarithmic geometry, we compute this cohomology group on the special fiber and then exploit its combinatorics to reconstruct the monodromy. As a result, we give a completely algebraic proof of the invariant cycles theorem for curves.
Title: Second Chern class of Fano manifolds and anti-canonical geometry
Abstract: Let X be a Fano manifold of Picard number one. We establish a lower bound for the second Chern class of X in terms of its index and degree. As an application, if Y is a n-dimensional Fano manifold with −KY=(n−3)H for some ample divisor H, we prove that h0(X,H)≥n−2. Moreover, we show that the rational map defined by |mH| is birational for m≥5, and the linear system |mH| is basepoint free for m≥7. As a by-product, we can obtain some similar results about singular weak Fano varieties of dimension at most four.
Title: Localizations and completions in motivic homotopy theory
Abstract: In the framework of motivic homotopy theory of Morel-Voevodsky, we will study some examples of homology localizations and nilpotent completions in the sense of Bousfield. In some easy but already interesting situation, we will answer to the following question: what happens if we take the Morel-Voevodsky stable category and formally invert those maps that induce an isomorphism in a fixed cohomology H*? We will give a very explicit answer in the case, for instance, where H* is motivic cohomology and we will discuss some further applications.
Title: Rational curves of low degree on cubic fourfolds and stability conditions
Abstract: A famous result of Beauville and Donagi states that the Fano variety of lines on a cubic fourfold is a smooth projective irreducible holomorphic symplectic (IHS) variety of dimension four, equivalent by deformation to the Hilbert square on a K3 surface. More recently, Lehn, Lehn, Sorger and van Straten constructed an IHS eightfold of K3 type from twisted cubic curves on a cubic fourfold Y non containing a plane.
In this talk, I will give an interpretation of the Fano variety of lines and of the LLSvS eightfold as moduli spaces of Bridgeland stable objects in the Kuznetsov component of Y. As a consequence, we reprove the categorical version of Torelli Theorem for cubic fourfolds, we obtain the identification of the period point of the LLSvS eightfold with that of the Fano
variety, and we discuss the derived Torelli Theorem for cubic fourfolds. This is a joint work with Chunyi Li and Xiaolei Zhao.
Title: Zero cycles on moduli spaces of stable curves
Abstract: Tautological zero cycles form a one-dimensional subspace of the set of all algebraic zero-cycles on the moduli space of stable curves. The full group of zero cycles can in general be infinite-dimensional, so not all points of the moduli space will represent a tautological class. In the talk, we will present geometric conditions ensuring that a pointed curve does define a tautological point. On the other hand, given any point Q in the moduli space we can find other points P_1, ..., P_m such that Q+P_1+ ... + P_m is tautological. The necessary number m is uniformly bounded in terms of g,n, but the question of its minimal value is open. This is joint work with R. Pandharipande.
Title: Moduli spaces of curves via Schottky spaces over Z
Abstract: The different theories of rigid analytic geometry give a recipe to construct the analytification of an algebraic variety defined over a complete non-archimedean field, analogous to the classical analytification of a complex algebraic variety. If one works with non-archimedean geometry in the sense of Berkovich, this recipe can be adapted to include varieties defined over rings of integers of number fields.
In this talk, I briefly explain how to do this via Poineau’s theory of Berkovich spaces over Z, and use this framework to construct and study moduli spaces of Schottky groups and Mumford curves. The main result of this new approach, obtained in collaboration with Jérôme Poineau, is the existence of universal Mumford curves over Z and their uniformization.
Title: The geometry of De Jonquières divisors on algebraic curves
Abstract: Enumerative geometry is an old subject with roots in the 19th century whose aim is to count the number of geometric objects of a certain type that satisfy given conditions. In this talk I will describe a classical enumerative problem, namely De Jonquières' count of certain prescribed hyperplane tangency conditions to a smooth curve embedded in projective space. I will explain how one may go about constructing the relevant moduli space for the problem and how one uses degenerations to singular curves to prove the validity of the counts.
Title: A variety that cannot be dominated by one that lifts.
Abstract: In the sixties, Serre constructed a smooth projective variety in characteristic p that cannot be lifted to characteristic 0. If a variety doesn't lift, a natural question is whether some variety related to it does lift. We construct an example of a smooth projective variety that cannot be rationally dominated by a smooth projective variety that lifts.