Algebraic Geometry Seminar Fall 2013

The seminar meets on Fridays at 2:25 pm in Van Vleck B231.

The schedule for the previous semester is here.

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Fall 2013 Schedule

date	speaker	title	host(s)
September 6	Matt Baker (Georgia Institute of Technology)	Metrized Complexes of Curves, Limit Linear Series, and Harmonic Morphisms	Melanie, Jordan
September 13	Nick Addington (Duke)	Hodge theory and derived categories of cubic fourfolds	Andrei
September 27	Parker Lowrey (UW-Madison)	TBA	Parker
October 4	Nathan Pflueger (Harvard)	Brill-Noether theory in low codimension	Melanie
October 25	Daniel Schultheis (University of Arizona)		Daniel
December 13	Vivek Shende (Berkeley)	Higher discriminants and the topology of algebraic maps	Melanie

Abstracts

Matt Baker

Metrized Complexes of Curves, Limit Linear Series, and Harmonic Morphisms

A metrized complex of algebraic curves is a finite edge-weighted graph G together with a collection of marked complete nonsingular algebraic curves C_v, one for each vertex; the marked points on C_v correspond to edges of G incident to v. We will present a Riemann-Roch theorem for metrized complexes of curves which generalizes both the classical and tropical Riemann-Roch theorems, together with a semicontinuity theorem for the behavior of the rank function under specialization of divisors from smooth curves to metrized complexes. As an application of the above considerations, we formulate a generalization of the notion of limit linear series to semistable curves which are not necessarily of compact type. This is joint work with Omid Amini. If time permits, we will also discuss how harmonic morphisms of metrized complexes can be used to provide a generalization of the Harris-Mumford theory of admissible coverings (joint work with Amini, Brugalle, and Rabinoff). This provides a "tropical" description of the tame fundamental group of an algebraic curve.

Nick Addington

Hodge theory and derived categories of cubic fourfolds

Cubic fourfolds behave in many ways like K3 surfaces. Certain cubics - conjecturally, the ones that are rational - have specific K3s associated to them geometrically. Hassett has studied cubics with K3s associated to them at the level of Hodge theory, and Kuznetsov has studied cubics with K3s associated to them at the level of derived categories.

These two notions of having an associated K3 should coincide. In joint work with Richard Thomas, we prove that they coincide generically: Hassett's cubics form a countable union of irreducible Noether-Lefschetz divisors in moduli space, and we show that Kuznetsov's cubics are a dense subset of these, forming a non-empty, Zariski open subset in each divisor.

Nathan Pflueger

Brill-Noether theory in low codimension

Abstract: 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 will 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. The proof relies on smoothing chains of elliptic curves, each joined to its neighbors at two points differing by a carefully chosen order of torsion.

Vivek Shende

Higher discriminants and the topology of algebraic maps

Abstract: We introduce `higher discriminants' of a morphism of complex algebraic varieties. These are defined in terms of transversality conditions, and we show: (1) the support of any summand of a projective pushforward of the IC sheaf is a component of a higher discriminant, and (2) any component of the characteristic cycle of a proper pushforward of the constant function is a conormal variety to a component of a higher discriminant.

This talk presents joint work with Luca Migliorini.