Moscow--Petersburg seminar on low-dimensional mathematics
SPb, PDMI, 29.03.2002, 16:00--17:45, room 311

Maxim Kazarian
Multisingularities and cobordisms

Enumerative geometry is traditionally considered as a
subject of intersection theory. Modern intersection theory with its
methods of singular schemes, blowing-ups, residue intersections, and
Hilbert schemes is rather technical and the proof of very
simple geometrically evident results require often many pages of
complicated formulas and commutative diagrams.

For example, the number of planes in $CP^3$ touching a fixed degree
$d$ smooth hypersurface at three points was computed by Salmon in the
middle of XIX-th century but the rigorous proof that would satisfy
modern algebraists has appeared comparatively recently. On the other
hand, the answer to similar question about hypersurfaces in $CP^4$,
$CP^5$,... the modern theory is unable to give.

In the talk we present an alternative  approach to enumerative
geometry whose motivation belongs to topology, more precisely, to the
theory of cobordisms and cohomological operations. We present several
hundred new enumerative formulas obtained with this approach which
solve, in particular, the following problem. 

Let $H\subset CP^n$ be a generic smooth hypersurface of degree $d$.
What is the number of $k$-dimensional subspaces $L$ such that the
intersection $L\cap H$ has prescribed collection
$\alpha_1,...,\alpha_r$ of isolated singularities? We present the answer
to this question for small $n,k,d$. For $n\le 3$ these formulas agree
with classical formulas of Plucker and Salmon.