UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

September 04, 2007

939 Evans Hall

3:45PM: Frobenius splittings and equations defining toric varieties

Sam Payne

If X in P^n is a smooth toric variety embedded in projective space, is X an isomorphic projection of an embedding in which X is cut out by quadrics? This notorious question remains wide open, and has generated a great deal of activity among toric geometers over the past fifteen years. Here I will discuss an approach via Frobenius splittings, a characteristic p technique that Mehta and Ramanathan introduced in the mid 1980s and that Ramanathan used to show the corresponding result for Schubert varieties. The majority of this talk will consist of a general introduction to Frobenius splitting techniques and a combinatorial interpretation of Frobenius splittings of toric varieties.

5:00PM: Free resolution of a module on a complete intersection

Ed Carter

For a regular local ring, a finite (and minimal) free resolution of the residue field is given by the Koszul complex. Over other local rings, however, the residue field has no finite free resolution. For instance, the minimal free resolution of the residue field k over the quotient ring k[x,y]_{(x,y)}/(x^2,y^2) is infinite. A general construction can be used to give the minimal free resolution of k over any quotient of k[x_1,...,x_n]_{(x_1,...,x_n)} by an ideal generated by a regular sequence. Over a complete intersection ring, when the rank of an infinite free resolution of a finitely generated module is bounded by some constant, the resolution must eventually become periodic of period 2. Every free resolution with period 2 also corresponds to a matrix factorization of some ring element x, where there are two matrices A and B such that the maps AB and BA are both multiplication by x on the appropriate free modules.

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