Fabio Tanturri (Genova)

Can a given polynomial (or some power thereof) be written as the determinant of a matrix with linear polynomials as entries? The answer to this question very much depends on the number of variables, the degree, and the generality of the polynomial. As it turns out, a positive answer is related to the existence of special vector bundles on the hypersurface cut out by the polynomial. In this talk I will try to explain this link and I will give an overview of the known results. In the second part, I will focus on two particular cases, cubic surfaces and cubic threefolds, and I will show how to explicitly realize their defining equation as the Pfaffian of a matrix with linear polynomials as entries.