Steven Kleiman, Ragni Piene, “Node Polynomials for Curves on Surfaces”, SIGMA, 2022, том 18,059, 23 стр.

Node Polynomials for Curves on Surfaces

Steven Kleiman^a, Ragni Piene^b

^a Room 2-172, Department of Mathematics, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
^b Department of Mathematics, University of Oslo, PO Box 1053, Blindern, NO-0316 Oslo, Norway

Аннотация: We complete the proof of a theorem we announced and partly proved in [Math. Nachr. 271 (2004), 69–90, math.AG/0111299]. The theorem concerns a family of curves on a family of surfaces. It has two parts. The first was proved in that paper. It describes a natural cycle that enumerates the curves in the family with precisely $r$ ordinary nodes. The second part is proved here. It asserts that, for $r\le 8$, the class of this cycle is given by a computable universal polynomial in the pushdowns to the parameter space of products of the Chern classes of the family.

Ключевые слова: enumerative geometry, nodal curves, nodal polynomials, Bell polynomials, Enriques diagrams, Hilbert schemes.

MSC: 14N10, 14C20, 14H40, 14K05

Поступила: 24 февраля 2022 г.; в окончательном варианте 28 июля 2022 г.; опубликована 2 августа 2022 г.

Язык публикации: английский

DOI: 10.3842/SIGMA.2022.059