Nodal deformations of hypersurfaces with an ordinary $m$-fold point

Abstract

Let $X\subset {\mathbb{O}}^n$ be a nodal hypersurface of degree $d$ with an ordinary $m$-fold point. Let ${\delta }_X$ be the largest value of $\delta$ such that $X$ is contained in the closure of the Severi variety of degree $d$ hypersurface in ${\mathbb{P}}^n$ with $\delta$ nodes. After recalling how the semicontinuity of the spectrum yields an upper bound for ${\delta }_X$, we produce various lower bounds for ${\delta }_X$. These bounds are given by polynomials in $m$ of degree $n$, with different leading coefficients. This improves previous bounds given by Ciliberto-Galati.