Polar syzygies in characteristic zero: The monomial case

Abstract

Given a set of forms f = { f 1 , … , f m } ⊂ R = k [ x 1 , … , x n ] , where k is a field of characteristic zero, we focus on the first syzygy module Z of the transposed Jacobian module D ( f ) , whose elements are called differential syzygies of f . There is a distinct submodule P ⊂ Z coming from the polynomial relations of f through its transposed Jacobian matrix, the elements of which are called polar syzygies of f . We say that f is polarizable if equality P = Z holds. This paper is concerned with the situation where f are monomials of degree 2, in which case one can naturally associate to them a graph G ( f ) with loops and translate the problem into a combinatorial one. The main result is a complete combinatorial characterization of polarizability in terms of special configurations in this graph. As a consequence, we show that polarizability implies normality of the subalgebra k [ f ] ⊂ R and that the converse holds provided the graph G ( f ) is free of certain degenerate configurations. One main combinatorial class of polarizability is the class of polymatroidal sets. We also prove that if the edge graph of G ( f ) has diameter at most 2 then f is polarizable. We establish a curious connection with birationality of rational maps defined by monomial quadrics.