# Mathematical Sciences Research Institute

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# Seminar

Commutative Algebra and Algebraic Geometry (Eisenbud Seminar) April 23, 2013 (03:45 PM PDT - 06:00 PM PDT)
Parent Program: -- UC Berkeley
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Commutative Algebra and Algebraic Geometry Seminar
Tuesdays, 3:45-6:00
939 Evans Hall

Organized by David Eisenbud

3:45 PM
Finite criteria for a ring to be Golod
Speaker: Luchezar Avramov
Abstract: Golod rings can be characterized by a numerical condition on the Betti numbers of the residue class field, or by a homological condition on the products in Koszul homology. The original proof of the equivalence shows that each condition can be checked in a known number of steps, but many subsequent proofs do not provide such information. The talk will explain a proof that does.

5 PM
Sharp upper bounds for higher linear syzygies of projective varieties
Speaker: Kangjin Han
Abstract: In this talk, we are going to consider upper bounds of higher linear syzygies i.e. graded Betti numbers in the first linear strand of the minimal free resolutions of projective varieties in arbitrary characteristic. For this purpose, we first remind Partial Elimination Ideals (PEIs)' theory and introduce a new framework in which one can study the syzygies of embedded projective schemes well using PEIs theory and the reduction method via inner projections.

Next we establish quite useful inqualities which govern the relations between the graded Betti numbers in the first linear strand of an algebraic set $X\subset\mathbb{P}^{N}$ and the ones of its inner projection $X_q\subset\mathbb{P}^{N-1}$. Using these results, we obtain some natural sharp upper bounds for higher linear syzygies of any nondegenerate projective variety in terms of the codimension with respect to its own embedding and classify what the extremal case and next to the extremal case are. This gives us interesting generalizations of classical characterizations on varieties of small degree by Castelnuovo and Fano from the viewpoint of syzygies'. Note that our method could be also applied to get similar results for more general categories (e.g. connected in codimension one algebraic sets). This is a joint work with Sijong Kwak.