Commutative Algebra and Polyhedra Seminar

Fridays at 2:35 PM
1234 LGRT

GAPS Fall 2005

One of the goals of this semester's seminar is to get a better understanding of Toric Duality, Bezoutians, and Tate resolutions. We will also discuss questions concerning the Castelnuovo-Mumford regularity. Most of the talks will be introductory and will not assume deep knowledge of algebraic geometry or commutative algebra.

Date	Title	Speaker
February 10	Organization Meeting
February 24	Toric Duality, Koszul Complexes and Spectral Sequences	Evgeny Materov
March 3	Gorenstein Duality and Bezoutians I	David Cox
March 10	Gorenstein Duality and Bezoutians II	David Cox
March 17	Gorenstein Duality and Bezoutians III	David Cox
March 24	No seminar. Spring break (aka reading week).
March 31	Gorenstein Duality and Bezoutians IV	David Cox
April 7	No seminar
April 14	Duality and Tate Resolutions	David Cox
April 21	Toric residues and Bezoutians	Ivan Soprunov
April 28	Dual Defect Toric Varieties	Eduardo Cattani
May 12	Dual Defect Toric Varieties II	Eduardo Cattani

Eduardo Cattani, UMass. "Dual Defect Toric Varieties"

Given a toric variety X_A, its dual X* is the Zariski closure of the locus of hyperplanes tangent to X_A at a smooth point. Generically, X* is a hypersurface and its equation, suitably normalized, is called the A-discriminant. If X* has higher codimension, then X_A is said to be a dual defect (toric) variety. I will show how restriction properties of the A-discriminant leads to a characterization of the Gale dual of a dual defect toric variety. From this characterization we can deduce a complete classification for low codimension. This is joint work with Ray Curran and a preprint is available here.

Ivan Soprunov, UMass. "Toric Residues and Bezoutians"

We have seen in the previous lectures that the Bezoutian gives duality, in particular, its "constant term" is a polynomail of critical degree, whose trace equals 1. In the toric setting the trace map is given by the toric residue map. I will explain when this map is not identically zero, and will provide a construction for the element of critical degree whose toric residue is 1. In the classical case it coincides with the constant term of the Bezoutian. The talk is based on a joint paper with Amit Khetan.

David Cox, Amherst College. "Duality and Tate Resolutions."

In this lecture, I will discuss some aspects of Tate resolutions and explain their conjectural relation to Bezoutians. My discussion of the Tate resolution will use the notation in the handout distributed on March 31. One warning is that my treatment of Tate resolutions will be far from complete -- other people will need to volunteer to give talks if we are to learn the full story.

David Cox, Amherst College. "Gorenstein Duality and Bezoutians IV."

In this talk, I will first complete the proof that the Bezoutian gives duality. Then I will discuss a slightly different kind of duality theorem which (conjecturally) is also described using Bezoutians. There will be one more talk in this series on April 14 where I will explain (again conjecturally) how Bezoutians relate to Tate resolutions.

David Cox, Amherst College. "Gorenstein Duality and Bezoutians III."

In this third talk on Gorenstein Duality and Bezoutians, I will prove a lemma about comparing regular sequences. This will complete the proof of duality. I will also discuss the role of the Jacobian and the extent to which the trace map I constructed is canonical.

David Cox, Amherst College. "Gorenstein Duality and Bezoutians II."

This is the second of two talks that explain the algebra involved in understanding the duality between the graded pieces of the quotient of a polynomial ring by a regular sequence of maximal length.

David Cox, Amherst College. "Gorenstein Duality and Bezoutians I."

Let S be a polynomial ring in n+1 variables and let I be an ideal in S generated by n+1 homogeneous polynomials. If the polynomials form a regular sequence, then we get a duality between the graded pieces of the quotient ring. Furthermore, this duality can be written down explicitly in terms of the Bezoutian of the polynomials generating I. My plan is to describe these results carefully, give examples, and indicate some major steps in the proof. I expect to need two lectures (March 3 and 10) to cover everything.

Evgeny Materov, UMass. "Toric Duality, Koszul Complexes and Spectral Sequences."

Let S be the homogeneous coordinate ring of a complete toric variety X. We study the duality between graded parts of the quotient ring R = S/I, where I is the ideal generated by dim(X) + 1 polynomials in S that don't vanish simultaneously on X. This duality is related to the theory of multivariable residues and also appears in the study of Frobenius algebras and multivariable Bezoutians. Unlike the classical situation for X = P^n, the pairing between the graded parts of R can be non-perfect which can be shown using the machinery of spectral sequences. In particular, I will illustrate the failure of the duality by some concrete examples when X is a product of projective spaces.