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 δ \delta dimensional linear space of degree d d real hypersurfaces containing 1 δ ! ( γ n d n ) δ + O ( d n δ − 1 ) \frac {1}{\delta !}(\gamma _nd^n)^{\delta }+O(d^{n\delta -1}) hypersurfaces with δ \delta real nodes, where γ n \gamma _n are positive and given by a recursive formula. This is asymptotically comparable to the number 1 δ ! ( ( n + 1 ) ( d − 1 ) n ) δ + O ( d n ( δ − 1 ) ) \frac {1}{\delta !} \left ( (n+1)(d-1)^n \right )^{\delta }+O\left (d^{n(\delta -1)} \right ) of complex hypersurfaces having δ \delta nodes in a δ \delta dimensional linear space. In the case δ = 1 \delta =1 we give a slightly better leading term.