Algebraic Geometry Seminar Fall 2013
The seminar meets on Fridays at 2:25 pm in Van Vleck B231.
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Fall 2013 Schedule
|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|
|October 4||Nathan Pflueger (Harvard)||Brill-Noether theory in low codimension||Melanie|
|October 11||Parker Lowrey (UW-Madison)||Grothendieck-Riemann-Roch for derived schemes||Parker|
|October 25||Daniel Schultheis (University of Arizona)||Daniel|
|November 8||Adrian Clingher (University of Missouri - St. Louis)||K3 Surfaces of High Picard Rank||Max|
|December 6||Roman Fedorov (Kansas State University)||Dima|
|December 13||Vivek Shende (Berkeley)||Higher discriminants and the topology of algebraic maps||Melanie|
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.
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.
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.
Grothendieck-Riemann-Roch for derived schemes
Abstract: The usefulness of the various Riemann-Roch formulas as computational tools is well documented in literature. Grothendieck-Riemann-Roch is a commutative diagram relating pull-back in K-theory to the pull-back 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.
K3 Surfaces of High Picard Rank
Abstract: I will discuss several special families of complex algebraic K3 surfaces of Picard rank 16 or higher. In terms of Hodge theory, these surfaces are related, via the Kuga-Satake correspondence, to certain abelian four-folds. The talk will outline the geometry of the correspondence, as well as present an explicit classification of these special K3 surfaces in terms of modular forms of appropriate type.
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.