Abstract
For a relative effective divisor mathcal {C} on a smooth projective family of surfaces q:mathcal {S}rightarrow B, we consider the locus in B over which the fibres of mathcal {C} are delta -nodal curves. We prove a conjecture by Kleiman and Piene on the universality of an enumerating cycle on this locus. We propose a bivariant class gamma (mathcal {C})in 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_*(c_1(mathcal {O}(mathcal {C}))^a c_1(T_{mathcal {S}/B})^b c_2(T_{mathcal {S}/B})^c). Under an ampleness assumption, we show that gamma (mathcal {C})cap [B] is the class of a natural effective cycle with support equal to the closure of the locus of delta -nodal curves. Finally, we apply our method to calculate node polynomials for plane curves intersecting general lines in mathbb {P}^3. We verify our results using nineteenth century geometry of Schubert.
Highlights
1.1 The Kleiman–Piene conjectureAll schemes we consider are separated and of finite type over C
Let B be a base scheme, and let q : S → B be a smooth family of surfaces, i.e. a smooth projective morphism of relative dimension 2
In this paper we propose a class γ (C) ∈ Aδ(B), enumerating the δ-nodal curves, inspired by the BPS calculus of Pandharipande and Thomas [26]
Summary
All schemes we consider are separated and of finite type over C. In [19] and [17], the authors prove the following theorem: Theorem 1.1 (Kleiman–Piene) Under the above hypotheses DIMKP, the locus B(δ) of δ-nodal curves is either empty, or has pure codimension δ. We will show that if B is complete, the class γ (C) ∩ [B] is the rational equivalence class of a natural cycle with support B(δ) For this we work with hypotheses DIM, similar to but slightly weaker than DIMKP, and an additional ampleness assumption AMP. By means of a family version of an algorithm by Ellingsrud, Göttsche and Lehn [8], we show that without assumptions, the class γ (C) is a universal polynomial in the classes (a, b, c) This will be the content of Theorem A below
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