Hadamard products and binomial ideals

Abstract

We study the Hadamard product of two varieties V and W, with particular attention to the situation when one or both of V and W is a binomial variety. The main result of this paper shows that when V and W are both binomial varieties, and the binomials that define V and W have the same binomial exponents, then the defining equations of V⋆W can be computed explicitly and directly from the defining equations of V and W. This result recovers known results about Hadamard products of binomial hypersurfaces and toric varieties. Moreover, as an application of our main result, we describe a relationship between the Hadamard product of the toric ideal IG of a graph G and the toric ideal IH of a subgraph H of G. We also derive results about algebraic invariants of Hadamard products: assuming V and W are binomial with the same exponents, we show that deg⁡(V⋆W)=deg⁡(V)=deg⁡(W) and dim⁡(V⋆W)=dim⁡(V)=dim⁡(W). Finally, given any (not necessarily binomial) projective variety V and a point p∈Pn∖V(x0x1⋯xn), subject to some additional minor hypotheses, we find an explicit binomial variety that describes all the points q that satisfy p⋆V=q⋆V.