Midwest
Algebra,
Geometry
and their Interactions
Conference
- MAGIC'10 -
University of Notre Dame,
Notre Dame April 23-25, 2010
Abstracts
Title:Kuga-Satake construction for K3 surfaces near
the boundary of the moduli
Valery Alexeev, University of Georgia Abstract:
Mumford observed that the Torelli map from Mg (moduli of curves) to Ag
(moduli of principally polarized abelian varieties), sending a smooth
curve of genus g to its Jacobian, can be extended to a morphism
between compactifications, the Deligne-Mumford compactification of Mg
and a particular toroidal compactification of Ag. Namikawa showed how
to make this construction geometric, and I explained how to make it
functorial.
I will show that in several examples the Kuga-Satake construction,
sending a K3 surface to an abelian variety, behaves in a similar way
near the boundary of the respective moduli spaces.
Title:Rees algebras and singularities of rational plane curves
David Cox, Ahmerst College Abstract:
Geometric modelers study parameterized curves and surfaces
in the plane and 3-space. Their method of "moving curves" and "moving
surfaces" led them to discover the polynomial relations defining the
Rees algebra of the ideal generated by the polynomials that give the
parametrization. At the time they did this, they had no idea what a
Rees algebra was. Meanwhile, the commutative algebra community had
studied for many years the Rees algebras of various types of ideals,
though not those that arise from geometric modeling. Recently, the
interests of these groups have converged, leading to new results and
new hard problems to think about. My lecture will describe joint work
with Kustin, Polini and Ulrich on the test case of rational plane
sextics and the special role played by double and triple points. K3
surfaces will make a brief appearance toward the end of the talk in
connection with my fantasy that they might help distinguish between
certain types of Rees algebras.
Title:Singularities of pairs
Lawrence Ein, University of Illinois at Chicago Abstract:
Let X be a smooth variety and Y be a closed subvariety of X. We discuss
certain invariants attached to the singulariteis of the pair (X, Y)
from birational geometry.
Title:Liaison and Groebner bases of pfaffian ideals
Elisa Gorla, Universität Basel, Switzerland Abstract:
Ideals generated by pfaffians are of interest both in algebra
and geometry, and in their study we see a beautiful interplay of
algebraic and geometric techniques. In this talk we discuss recent
results on the liaison class of ideals generated by pfaffians of mixed
size in a ladder of a generic skew-symmetric matrix. We call ladder
pfaffian variety a scheme whose saturated ideal belongs to this family.
We show that ladder pfaffian varieties can be obtained from a linear
variety by a sequence of ascending G-biliaisons. By combining the
liaison result with a simple Hilbert function computation, we argue that
the pfaffians are a Groebner basis for the ideal that they generate.
Title:Uniformity of rational points on curves
Joseph Harris, Harvard University Abstract:
Faltings' theorem says that there are only finitely many
solutions to a diophantine equation of genus 2 or more. This has
naturally led to speculation about possible extensions: how the
theorem might be generalized to higher dimensions, and how the number
of solutions behaves when we vary the coefficients of the problem, to
name two. In this talk we'll discuss some of these generalizations,
and describe a surprising logical connection between them.
Title:Continuous Closure
Mel Hochster, University of Mighigan Abstract:
The talk will present joint work with Neil Epstein.
Let R be the coordinate ring of a Zariski closed algebraic
set X in complex n-space. An element g of R is said to be
in the continuous closure of an ideal J of R if g is
a linear combination of elements of J with
coefficients in the ring of continuous (in the Euclidean
topology) complex-valued functions on X. This notion was
introduced by Holger Brenner. Brenner also defined an algebraic
notion called axes closure, which always contains the
continuous closure. Brenner proved that these two
closures agree for primary monomial ideals in polynomial
rings, and asked whether they agree in general.
The talk will discuss what is known about this
question. For an ideal with no embedded prime ideals,
the two closures discussed above agree in general for all
reduced finitely generated algebras over the complex numbers.
However, Neil Epstein and the speaker have shown recently that continuous
and axes closure can disagree, even for monomial ideals, if there are
embedded primes. A recent algebraic characterization of the closure of a
monomial ideal in the general case will be given. It remains an open question
whether the notion of continuous closure coincides with an algebraically
defined closure in the general case.
Title:Cohomology groups of structure sheaves
János Kollár, Princeton University Abstract:
I will discuss the behavior of cohomology groups
of the structure sheaf and of the dualizing sheaf under
deformations and birational maps.
Title:Positivity of cycles on abelian varieties
Robert Lazarsfeld, University of Michigan Abstract:
The cones of divisors and curves defined by various positivity
conditions on a smooth projective variety have been the subject of a
great deal of work in algebraic geometry, and by now they are quite
well understood. However the analogous cones for cycles of higher
codimension have started to come into focus only recently. I will
discuss a couple of computations on abelian varieties where one can
work out the picture fairly completely -- already here one sees some
non-classical phenomena. I will also discuss some of the many open
problems that present themselves around this circle of ideas. (This is
joint work in progress with Olivier Debarre, Lawrence Ein and Claire
Voisin.)
Title:Cohomological characterization of vector bundles
Rosa Maria Miró-Roig, Universitat de Barcelona, Spain Abstract:
In my talk, I will address the problem of giving a cohomological
characterization of vector bundles on algebraic varieties.
This is a longstanding problem in Algebraic geometry which has its roots
in an old paper by Horrocks where he gave a cohomological characterization
of line bundles on projective spaces Pn.
In my talk, I will give a cohomological characterization of the bundle of
p-differential forms on multiprojective spaces
Pn1 x ... x Pns
and a cohomological characterization of Steiner bundles on
algebraic varieties. As a main tool I will use a generalized version
of Beilinson's spectral sequence.
This is joint work with Costa and Soares.
Title:Recent work in Numerical Algebraic Geometry
Andrew Sommese, University of Notre Dame Abstract:
The talk will start with a brief overview of Numerical
algebraic geometry, the numerical computation and manipulation
of algebraic varieties.
There will be a discussion of two recent advances
regeneration (with J. Hauenstein and C. Wampler)
local dimension testing (with D. Bates, J. Hauenstein,
and C. Peterson).
The strong parallelism intrinsic to the current algorithms
combined with inexpensive clusters have allowed solution of
problems involving systems of polynomials that were until
recently beyond the pale. As illustrations of this we discuss
our ongoing work (with W. Hao, J. Hauenstein, B. Hu, Y. Liu,
and Y. Zhang) on the solution of systems of differential
equations coming from pattern formation on zebra fish and
different tumor growth models.
