2014/09/17 by Block, Florian, Göttsche, Lothar · 1 citation
#Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1409.4868
A convex lattice polygon Delta determines a pair (S,L) of a toric surface together with an ample toric line bundle on S. The Severi degree NDelta,delta is the number of delta-nodal curves in the complete linear system |L| passing through dim|L|-delta general points. Cooper and Pandharipande showed that in the case of PP1 x PP1 the Severi degrees can be computed as the matrix elements of an operator on a Fock space. In this note we want to generalize and extend this result in two ways. First we show that it holds more generally for Delta a so called h-transverse lattice polygon. This includes the case of PP2 and rational ruled surfaces, but also many other, also singular, surfaces. Using a deformed version of the Heisenberg algebra, we extend the result to the refined Severi degrees defined and studied by Göttsche and Shende and by Block and Göttsche. For Delta an h-transverse lattice polygon, one can, following Brugallé and Mikhalkin, replace the count of tropical curves by a count of marked floor diagrams, which are slightly simpler combinatorial objects. We show that these floor diagrams are the Feynman diagrams of certain operators on a Fock space, proving the result.