Polar syzygies in characteristic zero: the monomial case

Autores: María Isabel Bermejo Díaz , Philippe Giménez , Aron Simis
Localización: Journal of pure and applied algebra, ISSN 0022-4049, Vol. 213, Nº 1, 2009, págs. 1-21
Idioma: inglés
DOI: 10.1016/j.jpaa.2008.05.001
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Resumen
- Given a set of forms , where k is a field of characteristic zero, we focus on the first syzygy module of the transposed Jacobian module , whose elements are called differential syzygies of . There is a distinct submodule coming from the polynomial relations of through its transposed Jacobian matrix, the elements of which are called polar syzygies of . We say that is polarizable if equality holds. This paper is concerned with the situation where are monomials of degree 2, in which case one can naturally associate to them a graph 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 and that the converse holds provided the graph 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 has diameter at most 2 then is polarizable. We establish a curious connection with birationality of rational maps defined by monomial quadrics.