QLunch: Yeongrak Kim

Speaker: Yeongrak Kim, Pusan National University

Title: On Hessian matrix factorizations

Abstract: Let F be a polynomial. A pair of matrices (A, B) with polynomial entries is called a matrix factorization of F if AB=BA=F I_n for some n where I_n denotes the n×n identity matrix. Theory of matrix factorization, introduced by Eisenbud to study homological algebra over a hypersurface ring, becomes popular thanks to its various connections between (non-)commutative algebra, ACM/Ulrich sheaves, complexity theory, singularity theory, and mathematical physics. It is practically hard to achieve a matrix factorization of a given polynomial, and it seems to be extremely difficult to classify these matrix factorizations. One naive idea to obtain a sqaure matrix from a given polynomial F is by taking the Hessian matrix H of F, and in a very few cases H provides a matrix factorization of F itself, which I will call it a Hessian matrix factorization.

The simplest nontrival case is when F=xyz defined in three variables. In this case, the gradient of F defines so-called the standard Cremona transformation of the projective plane. I will talk about the complete classification on homogeneous cubic forms F having graded Hessian matrix factorizations. The key ingredients are the classfication of the EKP-homaloidal cubics and the XJC-correspondence introduced by Pirio and Russo. If time permits, I will show you a few examples on quartic forms having Hessian matrix factorizations.