The general quadruple point formula

TL;DR

This paper establishes a universal quadruple point formula for maps between manifolds, extending known relations to multisingularities beyond Morin-maps, with applications in enumerative geometry.

Contribution

It proves the first general quadruple point formula outside Morin-maps, advancing the understanding of multisingularity relations and their applications.

Findings

Derived the first general quadruple point formula for multisingularities.

Applied the formula to compute 4-secant linear spaces in projective varieties.

Explored formulas for higher tuple points and specific multisingularities.

Abstract

Maps between manifolds $M^m\to N^{m+\ell}$ ($\ell>0$) have multiple points, and more generally, multisingularities. The closure of the set of points where the map has a particular multisingularity is called the multisingularity locus. There are universal relations among the cohomology classes represented by multisingularity loci, and the characteristic classes of the manifolds. These relations include the celebrated Thom polynomials of monosingularities. For multisingularities, however, only the form of these relations is clear in general (due to Kazarian), the concrete polynomials occurring in the relations are much less known. In the present paper we prove the first general such relation outside the region of Morin-maps: the general quadruple point formula. We apply this formula in enumerative geometry by computing the number of 4-secant linear spaces to smooth projective varieties. Some other multisingularity formulas are also studied, namely 5, 6, 7 tuple point formulas, and one corresponding to $\Sigma^2\Sigma^0$ multisingularities.