Algebraic Geometry Seminar

 Department of Mathematics
Princeton University

Spring 2009 Lectures

Regular meeting time: Tuesdays 4:30-5:30 (Tea served at 3:30)
Place: Fine 322

Date Speaker Title
Feb 3 Michael Zieve
IAS		 Automorphism groups of curves
Hurwitz proved that a complex curve of genus g>1 has at most 84(g-1) automorphisms. In case equality holds, the automorphism group has a quite special structure. However, in a qualitative sense, all finite groups G behave the same way: the least g>1 for which G acts on a genus-g curve is on the order of (#G)*d(G), where d(G) is the minimal number of generators of G. I will present joint work with Bob Guralnick on the analogous question in positive characteristic. In this situation, certain special families of groups behave fundamentally differently from others. If we restrict to G-actions on curves with ordinary Jacobians, we obtain a precise description of the exceptional groups and curves.
Feb 10 Seminar will not be held today
 We are experiencing technical difficulties.
Feb 17 Yu Yasufuku
CUNY		Vojta's conjecture on Blowups and GCD Inequalities
Vojta's conjecture is a deep conjecture in Diophantine geometry, implying for example the Bombieri-Lang conjecture and the abc conjecture.  In this talk, I will show some cases of the conjecture for blowup varieties.  As a consequence, we derive some interesting inequalities of greatest common divisors.  An important ingredient will be Schmidt's subspace theorem, both directly and indirectly through the results of Corvaja and Zannier.
Feb 24 Yichao Tian
IAS
Canonical subgroup for p-divisible groups
 Let R be a discrete valuation ring of mixed characteristic with a perfect
residue field of characteristic p>0, and G be a p-divisible group over R. In this talk,
we are interested in a conjecture of Lubin on the existence of a canonical subgroup of G
which lifts the kernel of the Frobenius homomorphism of the special fiber of G. If G is
"not too supersingular", a condition expressed in terms of the p-adic valuation of the
determinant of a certain map attached to G,  we claim that a certain degree of 
Abbes-Saito's canonical filtration for the kernel of multiplication by p on G gives the
answer. In order to do this, we will introduce the Bloch-Kato filtration for a finite and
flat group scheme over R killed by p, and give also a description of its Abbes-Saito's
filtration in terms of congruence subgroups.
Mar 3 Burt Totaro
Cambridge
Algebraic surfaces and hyperbolic geometry
The intersection form on the group of line bundles on a complex algebraic surface always has signature (1,n) for some n.  So the automorphism group of an algebraic surface always acts on hyperbolic n-space. For a class of surfaces including K3 surfaces and many rational surfaces, there is a close connection between the properties of the variety and the corresponding group acting on hyperbolic space. (In fancier terms: the Morrison-Kawamata cone conjecture holds for klt Calabi-Yau pairs in dimension 2.)
Mar 10 Eduardo Esteves
IMPA		Compactified Jacobians and Abel maps for singular curves
We will discuss the problem of extending the construction of the classical Abel maps for smooth curves to the case of singular curves. The construction of degree-1 Abel maps will be shown, together with an approach for constructing higher degree Abel maps.
Mar 17 Spring BreakNo meeting
Mar 24 Junecue Suh
MIT		Nonhomeomorphic conjugates of connected Shimura varieties
Joint work with J. S. Milne. Start with a complex algebraic variety and make a conjugate of it by applying an automorphism of \C to its defining equations. While, by Grothendieck, the two conjugate varieties have the same algebraic fundamental group, Serre showed that the _topological_ fundamental groups can still be different. Other examples followed later.

In this talk, we construct connected Shimura varieties whose conjugates have nonisomorphic topological fundamental groups, by using Margulis's super-rigidity. In particular, we get examples that are moduli spaces of abelian varieties with prescribed (PEL) structure and Eilenberg-MacLane spaces at the same time.
Mar 31Nicolas Templier
IAS
Arakelov invariants on modular curves
Arakelov theory provides a rich set of invariants. We shall discuss the question of their limiting behavior in several classical examples, with an emphasis on heights of special points and of modular curves.
Apr 7 Jarod Alper
Columbia
Constructing moduli spaces of objects with infinite automorphisms
Moduli problems parameterizing objects with infinite automorphisms (eg. semi-stable vector bundles) often do not admit coarse moduli schemes but may admit moduli schemes identifying certain non-isomorphic objects.  I will introduce techniques to study such moduli stacks and address the question of how such moduli schemes can be intrinsically constructed.  The crucial ingredient is the notion of a good moduli space for an Artin stack, which generalizes Mumford's geometric invariant theory and characterizes the desired geometric properties of a moduli scheme parameterizing objects with infinite automorphisms.
Apr 14 Brian Osserman
UC Davis		 Vector bundles with sections
Classical Brill-Noether theory studies, for given g, r, d, the space of line bundles of degree d with r+1 global sections on a curve of genus g. We will review the main results in this theory, and the role of degeneration techniques in proving them, and then we will discuss the
situation for higher-rank vector bundles, where even the most basic questions remain wide open.
Apr 21 Bjorn Poonen
MIT		 
Automorphisms mapping a point into a subvariety
Given a variety X, a point x in X, and a subvariety Z of X, is there an automorphism of X mapping x into Z?
We prove that this problem is undecidable.
Apr 28,
5 PM
(note special time)		 Chad Schoen
Duke
 Calabi-Yau threefolds with vanishing third Betti number
Smooth, projective, three dimensional, algebraic varieties with trivial canonical sheaf and vanishing third etale Betti number do not exist over fields of characteristic zero. In the past few years a number of examples have been found in positive characteristic. Some of these examples and questions they raise will be discussed.


Other seminars in this department

For more information about this seminar, contact Alexei Oblomkov (oblomkov@math) and Max Lieblich (lieblich@math).