The F-signature of Determinantal Rings

Abstract

Let $R_{n,p}$ be the determinantal hypersurface ring defined by the determinant of a generic $n\times n$ matrix in characteristic $p$ where $n\ge 2$. In this paper, we obtain explicit bounds for the $F$-signature of $R_{n,p}$. For the upper bound, we prove that this $F$-signature is decreasing in $n$, so the $F$-signature of $R_{2,p}$ serves as an upper bound, which is $2/3$. For the lower bound, we find a toric degeneration $B_n$ of $R_{n,p}$, so the $F$-signature of $B_n$ serves as a lower bound. This $F$-signature can be computed using either Gröbner basis or Han-Monsky machinery, and its value is $\frac{(n!{)}^2}{(2n-1)!}$.