Hyperplane arrangements, resonance varieties, and the zero-divisor cup length
Nathan Fieldsteel (UIUC Math)
Abstract: For an $R$-algebra $A$, the zero-divisor cup length of $A$ is the maximum length of a non-zero product in the ideal of zero divisors in $A \otimes_R A$. We are interested in the zero-divisor cup length in the case where $A$ is the Orlik-Solomon algebra of a hyperplane arrangement. For generic arrangements we would like to confirm a conjecture, due to Yuzvinsky, that gives a formula for the zero-divisor cup length. For non-generic arrangements, we would like a geometric or combinatorial predictor of when the zero-divisor cup length is "lower than expected", hopefully in terms of the resonance varieties of the arrangement. We will demonstrate the utility of Macaulay2 in performing computations related to these questions.