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Marian, Alina; Oprea, Dragos; Pandharipande, Rahul

Higher rank Segre integrals over the Hilbert scheme of points. (English) Zbl 1495.14006

J. Eur. Math. Soc. (JEMS) 24, No. 8, 2979-3015 (2022).
The determination of Segre classes of tautological vector bundles on the Hilbert scheme of points on surfaces is a central problem in enumerative geometry. More precisely, given a nonsingular projective surface \(S\) a vector bundle \(V\) on \(S\) induces naturally a tautological vector bundle \(V^{[n]}\) on the Hilbert scheme of points \(S^{[n]}\). The associated Segre series \begin{align*} S_{\alpha}(z)=\sum_{n=0}^{\infty}z^n\int_{S^{[n]}} s(V^{[n]}) \end{align*} was studied in [Invent. Math. 136, No. 1, 157–207 (1999; Zbl 0919.14001)] by M. Lehn and has been the object of intense research in the last decades. More in general similar series can be studied for a \(K\)-theory class \(\alpha\in K(S)\) and the induced class \(\alpha^{[n]}\in K(S^{[n]})\).
In particular G. Ellingsrud et al. [J. Algebr. Geom. 10, No. 1, 81–100 (2001; Zbl 0976.14002)] show a factorization \begin{align*} S_{\alpha}(z)=A_0(z)^{c_2(\alpha)} \cdot A_1(z)^{c_1(\alpha)^2} \cdot A_2(z)^{\chi(\mathcal O_S)} \cdot A_3(z)^{c_1(\alpha)\cdot K_S}\cdot A_4(z)^{K_S^2} \end{align*} in universal series \(A_0(z),\dots,A_4(z)\in \mathbb Q[[z]]\) depending on \(\alpha\) just through the rank.
The main results of the paper under review prove formulas for:
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\(A_0, A_1, A_2\) for \(S\) a \(K\)-trivial surface (and hence the factors involving \(A_3\) and \(A_4\) are trivial);
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\(A_3, A_4\) for any surface \(S\) when the rank of \(\alpha\) is 2.

The formulas for the \(K\)-trivial surfaces are obtained by finding an optimal geometric setup where to compute them. Whereas the formulas for \(A_3\) and \(A_4\) are determined by studying the blow-up of a \(K3\) surface at a point. Key ingredients in the proofs are Reider techniques and intersection excess formula.
Reviewer: Luca Giovenzana (Chemnitz)

Cited in 2 Reviews
Cited in 10 Documents

MSC:

14C05 Parametrization (Chow and Hilbert schemes)
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14J10 Families, moduli, classification: algebraic theory
14J28 \(K3\) surfaces and Enriques surfaces

Keywords:

Hilbert scheme of points; tautological integrals; Segre and Verlinde numbers

Citations:

Zbl 0919.14001; Zbl 0976.14002
PDFBibTeX XMLCite
\textit{A. Marian} et al., J. Eur. Math. Soc. (JEMS) 24, No. 8, 2979--3015 (2022; Zbl 1495.14006)
Full Text: DOI arXiv

Online Encyclopedia of Integer Sequences:

Expansion of the unique real solution y(t) to the equation y*(1+y)^2/((1-y)*(1-y^3)) = t/(1+3t) with initial value y(0)=0.

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