Spring 2015


Spring 2015 BC-Northeastern Algebraic Geometry Conference

Organized by:
Ana-Maria Castravet, Dawei Chen, Maksym Fedorchuk, Brian Lehmann, Emanuele Macri, and Alina Marian

When:
April 25, 2015, 10am-5pm

Where:
315 Shillman Hall, Northeastern University

Registration: Please register so that we have a headcount for coffee breaks.

Speakers:
Titles and schedule (subject to minor changes):

 10-11am
 
 11-11:30am
 
 11:30am-12:30pm


 2-3pm


 3-3:30pm
 
 3:30-4:30pm

 Yuri Tschinkel: K3 surfaces and their punctual Hilbert schemes
 
 Coffee break
 
 Paolo Stellari: Stability Conditions on Abelian and some Calabi-Yau Threefolds


 Brian Lehmann:
Zariski decompositions for curve classes


 Coffee break

 Filippo Viviani: Fourier-Mukai and autoduality for compactified Jacobians


Abstracts:

Brian Lehmann: Zariski decompositions for curve classes
Abstract: The Zariski decomposition is an important tool for understanding the sections of a divisor on a surface.  We discuss a generalization to curves on varieties of arbitrary dimension.  We give a number of examples and applications to birational geometry.

Paolo Stellari: Stability Conditions on Abelian and some Calabi-Yau Threefolds
Abstract: We prove that a Bogomolov-Gieseker type inequality holds for abelian threefolds and some smooth projective Calabi-Yau threefolds of quotient type. This was conjectured by Bayer, Macri' and Toda. From this one deduces that the corresponding spaces of Bridgeland stability conditions are not empty. This is joint work with A. Bayer and E. Macri'.

Yuri Tschinkel: K3 surfaces and their punctual Hilbert schemes
Abstract: I will talk about birational geometry of holomorphic symplectic varieties arising as deformations of punctual Hilbert schemes of K3 surfaces (joint with Brendan Hassett).

Filippo Viviani: Fourier-Mukai and autoduality for compactified Jacobians
Abstract: To every reduced (projective) curve X with planar singularities one can associate many fine compactified Jacobians, depending on the choice of a polarization on X, which are birational (possibly non-isomorphic) singular Calabi-Yau projective varieties, each of which yields a modular compactification of a disjoint union of copies of the generalized Jacobian of X. We define a Poincar\'e sheaf on the product of any two (possibly equal) fine compactified Jacobians of X and show that the associated integral transform is an equivalence of their derived categories, hence it defines a Fourier-Mukai transform. This generalizes the classical result of S. Mukai for Jacobians of smooth curves and the more recent result of D. Arinkin for compactified Jacobians of integral curves with planar singularities, and it provides further evidence  for the classical limit of the geometric Langlands conjecture (as formulated by Donagi and Pantev). As a corollary, we prove that there is a canonical isomorphism (called autoduality) between the generalized Jacobian of X and the connected component of the identity of the Picard scheme of any fine compactified Jacobian of X and that algebraic equivalence and numerical equivalence coincide on any fine compactified Jacobian.
This is a joint work with M. Melo and A. Rapagnetta.