Polar syzygies in characteristic zero: the monomial case

Given a set of forms f = {f1, . . . , fm} ⊂ R = k[x1, . . . , xn] of degree 2, 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, in which case one can naturally associate to them a graph G(f) with loops and translate the problem into a combinatorial one. A 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.