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

RESUMO

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.