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The general quadruple point formula
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 132, Number 4, August 2010
- pp. 867-896
- 10.1353/ajm.0.0130
- Article
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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.