##### Department of Mathematics,

University of California San Diego

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### Algebraic Geometry Seminar

## Jie Wang

#### UCSD

## Generic vanishing of Koszul cohomology: some recent progress

##### Abstract:

A central problem in curve theory is to describe the extrinsic geometry of algebraic curves in a given projective space with fixed genus and degree. Koszul cohomology groups in some sense carry everything one ever wants to know about the defining equations of a curve $X$ in $\mathbb{P}^r$: the number of independent equations of each degree vanishing on $X$ , the relations between the generators of the ideal $I_X$ of $X$, etc. In this talk, I will describe an inductive approach to study Koszul cohomology groups of general curves. In particular, we show that to prove the Maximal Rank Conjecture (for quadrics), it suffices to check all cases with the Brill-Noether number $\rho=0$. As a consequence, the Maximal Rank Conjecture holds if the embedding line bundles $L$ on $X$ satisfies the condition $h^1(L)<3$.

### October 31, 2014

### 2:30 PM

### AP&M 7218

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