On the Hadamard product of degenerate subvarieties

We consider generic degenerate subvarieties $X_i\subset\mathbb{P}^n$. We determine an integer $N$ and for $n\geq N$ 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 $nHadamard product can be singular and we give a lower bound for the dimension of the singular locus.