Algebraic Geometry Seminar Fall 2013
The seminar meets on Fridays at 2:25 pm in Van Vleck B219.
The schedule for the previous semester is here.
Algebraic Geometry Mailing List
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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)||Melanie|
|October 25||Daniel Schultheis (University of Arizona)||Daniel|
|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.
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.