Poster SessionSaturday, October 26, 4-7 pm, McGuinn Hall Lobby. Chih-Chi Chou (UIC)A vanishing theorem for log canonical pairs.We prove a Nadel vanishing type theorem for log canonical pairs. It partially generalizes a vanishing result due to de Fernex-Ein. We apply it to questions related to Castelnuovo-Mumford regularity of subvarieties of Schubert varieties. Stephen Coughlan (UMass Amherst)Derived categories of some fake del Pezzo surfaces.We describe a method for producing exceptional collections of line bundles on lots of surfaces of general type with geometric genus zero. Unlike genuine del Pezzo surfaces, such exceptional collections are never full, and give rise to mysterious phantom categories. Patricio Gallardo (Stony Brook)On the moduli space of quintic surfaces. We describe the use of GIT and stable replacement for studying the geometry of a special compactification of the moduli space of smooth quintic surfaces, the KSBA compactification. In particular we discuss the interplay between non-log-canonical singularities and boundary divisors. This interplay generalizes similar phenomena on the moduli space of curves of genus three. Jose Gonzalez (UBC)Operational Algebraic Cobordism.We associate a bivariant theory to any suitable oriented Borel-Moore homology theory on the category of algebraic schemes or the category of algebraic G-schemes. Applying this to the theory of algebraic cobordism yields operational cobordism rings and operational G-equivariant cobordism rings associated to all schemes in these categories. In the case of toric varieties, the operational T-equivariant cobordism ring may be described as the ring of piecewise graded power series on the fan with coefficients in the Lazard ring. These results are from joint work with Kalle Karu. Nathan Grieve (McGill University)Index conditions and cup-product maps on abelian varieties.We study questions surrounding cup-product maps which arise from pairs of non-degenerate line bundles on an abelian variety. Important to our work is Mumford's index theorem which we use to prove that non-degenerate line bundles exhibit positivity analogous to that of ample line bundles. As an application we determine the asymptotic behaviour of families of cup-product maps and prove that vector bundles associated to these families are asymptotically globally generated. John Lesieutre (MIT) Counterexamples to some positivity questions.There are several examples of varieties admitting infinite sequences of flops, including blow-ups of $\mathbb P^3$ at eight or more very general points and some Calabi-Yau threefolds. These sequences of flops can be exploited to construct counterexamples to several questions about positivity of divisors: nefness is not an open condition in families; the diminished base locus of a divisor is not always a closed set; Zariski decompositions do not necessarily exist in dimension three; asymptotic multiplicity invariants are not always finite in the relative setting; and the number of Fourier-Mukai partners of a variety can be infinite. Zhan Li (Rutgers University) Birationality of toric double mirrors.We prove that saturated generic complete intersections associated to double mirror nef-partitions are all birational. This result partially answers a question asked by Batyrev and Nill. Parker Lowrey (University of Wisconsin - Madison)Grothendieck-Riemann-Roch for derived schemes.The usefulness of the various Riemann-Roch formulas as computational tools is well documented in literature. Grothendieck-Riemann-Roch is a commutative diagram relating push-forward in K-theory to the push-forward of associated Chow invariants for locally complete intersection (l.c.i.) morphisms. We extend this notion to quasi-smooth morphisms between derived schemes, this is the "derived" analog of l.c.i. morphisms and it encompasses relative perfect obstruction theories. We will concentrate on the naturality of the construction from the standpoint of pure intersection theory and how it interacts with the virtual Gysin homomorphism defined by Behrend-Fantechi. Time permitting we will discuss the relationship with existing formulas, i.e., Ciocan-Fonanine, Kapranov, Fantechi, and Goettsche. Cesar Lozano Huerta (University of Illinois at Chicago)The Birational Geometry of Complete Quadrics.Schubert in the 19th century introduced the space of complete quadrics and it has attracted the attentions of geometers since. The purpose of the present poster is to show recent work on the birational geometry of this classical space. For example, we show the cone of effective divisors of the space of complete quadric surfaces has eight Mori chambers. Furthermore, we describe the moduli interpretation for all the minimal models induced by them. Some results and work in progress in the higher dimensional case are also presented. Sophie Marques (NYU CIMS) Tameness in arithmetic geometry and existence of slices.We are interested about ramification theory for actions involving group schemes. We compare two different possible definitions of tameness. We will present how the fundamental results of ramification theory in the algebraic number theory can be translated in this context and when we were able to prove them for action involving group schemes. We get for instance a result describing the structure of the inertia group for a tame action. This permits us to induce a tame action by an action of some extension of an inertia group. Jesus Martinez-Garcia (Johns Hopkins University) Kahler-Einstein metrics and birational geometry.Deciding when a Fano manifold admits a Kahler-Einstein metric is a complicated problem. It's been recently proved by Chen, Donaldson and Sun, and independently by Tian, that a Fano manifold admits a Kahler-Einstein metric if and only if it is K-stable. This is an algebraic concept that is complicated to compute. On the other hand, for those Fano manifolds which do not admit a Kahler-Einstein metric, we can approximate singular metrics to smooth ones. In this poster I explain how I have showed that those Del Pezzo surfaces which do not admit Kahler-Einstein metrics, do admit Kahler-Einstein metrics with edge singularities of small angles. The method used is a log version of Tian's alpha-invarant due to Jeffres, Mazzeo and Rubinstein, interpreted as a birational invariant known as the global log canonical threshold. Andrew Niles (UC Berkeley) Moduli of elliptic curves via twisted stable maps.I investigate compactifications of moduli stacks of elliptic curves with level structure arising as a consequence of the work of Abramovich, Olsson and Vistoli regarding twisted stable maps. I show that in fact these compactifications are precisely the Katz-Mazur regular models. Moreover, these stacks have natural moduli interpretations in terms of twisted curves, and I show how this relates to Drinfeld structures on generalized elliptic curves. Morgan Opie (UMass Amherst) Extremal divisors on moduli spaces of rational curves with marked points.We study effective divisors on $\overline{M}_{0,n}$, focusing on hypertree divisors introduced by Castravet-Tevelev and the proper transforms of divisors on $\overline{M}_{1,n-2}$ introduced by Chen-Coskun. We investigate and relate these two types of divisors, and from this construct extremal divisors on $\overline{M}_{0,n}$ for $n$ greater than or equal to $7$ that furnish counterexamples to the conjectural description of the effective cone of $\overline{M}_{0,n}$ given by Castravet-Tevelev. Nathan Pflueger (Harvard) Brill-Noether theory in low codimension.Brill-Noether theory studies the existence and deformations of curves in projective spaces; its basic object of study is W(g,r,d), the moduli space of smooth projective genus g curves with a choice of degree d line bundle having at least (r+1) independent global sections. The geometry of W(g,r,d) is depends on the number ρ = g-(r+1)(g-d+r). The Brill-Noether theorem, proved by Griffiths and Harris, states that when ρ is nonnegative, the map from W(g,r,d) to M_g is surjective, and a general fiber has dimension ρ. One may naturally conjecture that for ρ<0, W(g,r,d) is finite over a locus of codimension -ρ in M_g. This conjecture fails, but seemingly only when ρ is large compared to g. I discuss a proof that this conjecture holds for at least one component of W(g,r,d) in cases where 0 < -ρ < r/(r+2) g - 3r. Adriana Salerno (Bates College) Hypergeometric functions and the Dwork family.This is an algorithmic approach to computing the Gauss-Manin connection and Picard Fuchs equation associated to a family of hypersurfaces known most commonly as the Dwork family. We also study the connection between these results and hypergeometric functions. Taro Sano (University of Warwick, Princeton University)Deformations of Q-Fano 3-folds.Fano 3-folds with terminal singularities are important in the classification of algebraic 3-folds. However its classification is not done. I consider the existence of deformation of such Fano 3-folds which make the singularities milder. Benjamin Schmidt (Ohio State)
A generalized Bogomolov-Gieseker inequality for the smooth quadric threefold.Bridgeland stability conditions are an adaption of the concept of slope stability to triangulated categories. The main result on this poster is a certain generalized Bogomolov-Gieseker inequality as conjectured by Bayer, Macrì and Toda for the case of the smooth quadric threefold. This implies the existence of a family of Bridgeland stability conditions. Lei Song (UIC) On the singularities of effective locus of line bundles.We show that for a smooth complex projective variety $X$, if all divisors $D$ in the linear system $|L|$ are semi-regular, then the effective locus of line bundles in the Picard scheme has rational singularities around the point corresponding to $L$. Yu-Chao Tu (Princeton University)Gaffney-Lazarsfeld theorem for homogeneous space.We generalize the Gaffney-Lazarsfeld theorem on higher ramification loci of a branched covering of Pn to a homogeneous space with Picard number 1. As an application, we show that an irreducible normal projective variety having low degree branched covering to a homogeneous space with Picard number 1 is simply connected. Martin Ulirsch (Brown University)Tropical compactification in log-regular varieties.We define a natural tropicalization procedure for closed subsets of log-regular varieties and study its basic properties. This framework allows us to generalize some of Tevelev's results on tropical compactification to log-regular varieties. Jie Wang (University of Georgia) The primitive cohomology of the theta divisor of an abelian fivefold.The primitive cohomology of the theta divisor of a principally polarized abelian variety of dimension $g$ is a Hodge structure of level $g-3$. The Hodge conjecture predicts that it is contained in the image, under the Abel-Jacobi map, of the cohomology of a family of curves in the theta divisor. I will explain how one can use the Prym map to show that this version of the Hodge conjecture is true for the theta divisor of a general abelian fivefold. This is joint work with Izadi and Tam\'as. Hang Xue (Columbia University)The arithmetic and geometry of genus four curves.For a non-hyperelliptic genus four curve, we construct a canonical point in its Jacobian and study its properties. In particular, we show that it is non-torsion in the generic situation and its Neron--Tate height has the Northcott property if the base field is a number field. Xin Zhou (University of Michigan)Asymptotic Schur Decomposition of Veronese Syzygy Functors.The syzygies of Veronese embeddings are functors in the underlying vector spaces. We study the asymptotics of numerics of the linear strand as Schur functors. |

Boston College 2013 >