2017/10/05 by Ties Laarakker, Laarakker, Ties
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Polynomial and algebraic computation #Advanced Differential Equations and Dynamical Systems
paper · pdf · doi:10.48550/arxiv.1710.02085
For a relative effective divisor C on a smooth projective family of surfaces q:S→ B, we consider the locus in B over which the fibres of C are δ-nodal curves. We prove a conjecture by Kleiman and Piene on the univerality of an enumerating cycle on this locus. We propose a bivariant class γ(C)∈ A^*(B) motivated by the BPS calculus of Pandharipande and Thomas, and show that it can be expressed universally as a polynomial in classes of the form q_*(c1(O(C))a c1(TS/B)b c2(TS/B)c). Under an ampleness assumption, we show that γ(C)∩[B] is the class of a natural effective cycle with support equal to the closure of the locus of δ-nodal curves. Finally, we will apply our method to calculate node polynomials for plane curves intersecting general lines in ℙ3. We verify our results using 19th century geometry of Schubert.