The Hilbert function of some Hadamard products
The Hadamard product \(X*Y\)of two projective varieties \(X,Y\subset\mathbb P^n\) is the closure of the image of the rational map which sends \((x_0,\dots,x_n)\times (y_0,\dots,y_n)\in X\times Y\) to \((x_0y_0,\dots,x_ny_n)\). In geometric terms, it corresponds to a very special projection of the Segre product. Several applications of the Hadamard product to the statistics of Boltzmann machines and to matrix analysis suggest that a deeper knowledge of the geometric properties of \(X*Y\) can have relevant impact in applied algebraic geometry. In the paper under review, the authors study the Hilbert function (\(HF\)) of \(X*Y\). The main result says that \(HF(X*Y)\) is equal to the product \(HF(X)\cdot HF(Y)\) when either both \(X,Y\) are linear subvarieties, or when both \(X,Y\) have dimension at most \(1\).
scientific article; zbMATH DE number 6872991