Plucker-Clebsch formula in higher dimension

TL;DR

This paper establishes an upper bound on the number of double points in a general projection of a smooth surface in projective space, characterizing when equality occurs, and discusses extensions to higher dimensions.

Contribution

It provides a new inequality for the double point count of surfaces in projective space and characterizes the case of equality, extending classical formulas to higher dimensions.

Findings

The double point count $\delta_S$ is at most $inom{d-2}{2}$.

Equality holds if and only if the surface is a rational scroll.

Extensions to higher-dimensional varieties are considered.

Abstract

Let $S\subset\Ps^r$ ($r\geq 5$) be a nondegenerate, irreducible, smooth, complex, projective surface of degree $d$. Let $\delta_S$ be the number of double points of a general projection of $S$ to $\Ps^4$. In the present paper we prove that $ \delta_S\leq{\binom {d-2} {2}}$, with equality if and only if $S$ is a rational scroll. Extensions to higher dimensions are discussed.