UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

February 27, 2007

939 Evans


3:45: Janet's algorithm for modules over polynomial rings

Daniel Robertz

This talk gives an introduction to Janet bases.

Originally developed for the algebraic analysis of systems of partial differential equations in the beginning of the 20th century, the algorithm by Maurice Janet is today an efficient alternative for Buchberger's algorithm to compute Groebner bases of modules over polynomial rings.

In this talk we give a modern description of Janet's algorithm and explain nice combinatorial properties of the resulting Janet bases: separation of the variables into multiplicative and non-multiplicative ones for each Janet basis element allows to read off vector space bases for both the submodule and the residue class module. As a consequence, the Hilbert series and polynomial of a (graded) module as well as a free resolution are easily obtained from the Janet basis.

If time admits, some modifications of Janet's algorithm will be addressed which allow to work with polynomial rings over the integers instead of a field resp. generalize the algorithm to certain classes of non-commutative polynomial rings. Implementations are available in Maple and C++ and can also be demonstrated.

5:00: On the structure of certain natural sheaves in Gromov-Witten theory

Aleksey Zinger

A line bundle (or any sheaf) on Pn induces a sheaf over the moduli space of stable maps into Pn. In the genus-zero case, such sheaves are locally free and their euler classes encode GW-invariants of hypersurface. This was the starting point for the proof of the mirror symmetry for rational curves in the quintic threefold. In higher genera, these sheaves are not locally free and thus may not even have an euler class.

I will explain why such sheaves should come up the GW-theory of hypersurfaces. I will then present a geometric construction of the euler class of the restriction of such a sheaf to the main component of the moduli space (a generic element of which parametrizes a smooth curve) in the genus-one case. In a separate work with Jun Li, it is shown that this euler class encodes the "genus-one part" of the genus-one GW-invariant of a hypersurface. The euler class can be computed using a desingularization of the sheaf constructed with Ravi Vakil and Atiyah-Bott localization theorem.