Estimates of the number of singular points of a complex hypersurface and related questions

O. A. Ivanov   ; N. Yu. Netsvetaev

Zapiski Nauchnykh Seminarov POMI, Investigations in topology. Part 8, Tome 231 (1995), pp. 180-190

Résumé

It is well known that the number of isolated singular points of a hypersurface of degree $d$ in $\mathbb CP^m$ does not exceed the Arnol'd number $A_m(d)$, which is defined in combinatorial terms. In the paper it is proved that if $b^\pm_{m-1}(d)$ are the inertia indices of the intersection form of a nonsingular hypersurface of degree $d$ in $\mathbb CP^m$, then the inequality $A_m(d)<\min\{b^+_{m-1}(d),b^-_{m-1}(d)\}$ holds if and only if $(m-5)(d-2)\ge18$ and $(m,d)\ne(7,12)$. The table of the Arnol'd numbers for $3\le m\le14$, $3\le d\le17$ and for $3\le m\le8$, $d=18,19$ is given. Bibl. 6 titles.