Enumeration of algebraic and tropical singular hypersurfaces

Abstract

We develop a version of Mikhalkin's lattice path algorithm for projective hypersurfaces of arbitrary degree and dimension, which enumerates singular tropical hypersurfaces passing through appropriate configuration of points. By proving a correspondence theorem combined with the lattice path algorithm, we construct a $\delta$ dimensional linear space of degree $d$ real hypersurfaces containing $\frac{1}{\delta !}({\gamma }_n{d}^n{)}^{\delta }+O(d^{n\delta -1})$ hypersurfaces with $\delta$ real nodes, where ${\gamma }_n$ are positive and given by a recursive formula. This is asymptotically comparable to the number $\frac{1}{\delta !}{\left((n+1)(d-1{)}^n\right)}^{\delta }+O\left(d^{n(\delta -1)}\right)$ of complex hypersurfaces having $\delta$ nodes in a $\delta$ dimensional linear space. In the case $\delta =1$ we give a slightly better leading term.

48 pages, 5 figures; This is an edited version of my master thesis