Polar syzygies in characteristic zero: The monomial case

Abstract

a b s t r a c tGiven 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.

First Nonlinear Syzygies of Ideals Associated to Graphs

“…One can now determine the first row of the Betti diagram using the formulation of (2) given in [11, Proposition 2.1] when j = i + 2:…”

“…As a general aim, one would like to establish a correspondence between algebraic properties of the ideal I (or the ring R/I, or the subalgebra K[f ] ⊂ R) and the graph theoretical data of G(I) (or of any other graph associated to I such as G(I) c ). Two illustrations of this correspondence are the combinatorial characterizations of normality and polarizability of the subalgebra K[f ] ⊂ R given in [12] and [2], respectively. In this note, we shall focus on properties of I related to its minimal graded free resolution (1) 0…”

Syzygies of differentials of forms

“…We close with some general questions. In [1] the answer to the first of these questions is affirmative when f are monomials, hence so is the answer to the second question under the same assumption.…”

“…Then the presentation ideal of k[f ] is a codimension 3 almost complete intersection in which the quadrics generate a maximal regular sequence, while the fourth generator lives in degree 3. It can be shown that f is polarizable and that D is a free module generated by the differentials of the three quadrics (see [1,Example 5.21]).…”

Cremona maps defined by monomials

“…Thus, the initial step could be vacuous or, alternatively, would start from n = s 1 + 2, while in the inductive step one would then assume that n > s 1 + 2. Taking up the second alternative, the initial step has F = {x 2 1 , x 1 x 2 , . .…”