On the Hadamard product of degenerate subvarieties

Abstract

We consider generic degenerate subvarieties $X_i\subset {\mathbb{P}}^n$. We determine an integer $N$, depending on the varieties, and for $n\ge N$ we compute dimension and degree formulas for the Hadamard product of the varieties $X_i$. Moreover, if the varieties $X_i$ are smooth, their Hadamard product is smooth too. For $n<N$, if the $X_i$ are generically $d_i$-parameterized, the dimension and degree formulas still hold. However, the Hadamard product can be singular and we give a lower bound for the dimension of the singular locus.