## Geometry & Topology

### Universal polynomials for tautological integrals on Hilbert schemes

Jørgen Rennemo

#### Abstract

We show that tautological integrals on Hilbert schemes of points can be written in terms of universal polynomials in Chern numbers. The results hold in all dimensions, though they strengthen known results even for surfaces by allowing integrals over arbitrary “geometric” subsets (and their Chern–Schwartz–MacPherson classes).

We apply this to enumerative questions, proving a generalised Göttsche conjecture for all isolated singularity types and in all dimensions. So if $L$ is a sufficiently ample line bundle on a smooth variety $X$, in a general subsystem $ℙd ⊂|L|$ of appropriate dimension the number of hypersurfaces with given isolated singularity types is a polynomial in the Chern numbers of $(X,L)$.

When $X$ is a surface, we get similar results for the locus of curves with fixed “BPS spectrum” in the sense of stable pairs theory.

#### Article information

Source
Geom. Topol., Volume 21, Number 1 (2017), 253-314.

Dates
Revised: 15 December 2015
Accepted: 15 January 2016
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.gt/1510859134

Digital Object Identifier
doi:10.2140/gt.2017.21.253

Mathematical Reviews number (MathSciNet)
MR3608714

Zentralblatt MATH identifier
06687807

#### Citation

Rennemo, Jørgen. Universal polynomials for tautological integrals on Hilbert schemes. Geom. Topol. 21 (2017), no. 1, 253--314. doi:10.2140/gt.2017.21.253. https://projecteuclid.org/euclid.gt/1510859134

#### References

• P Aluffi, Euler characteristics of general linear sections and polynomial Chern classes, Rend. Circ. Mat. Palermo 62 (2013) 3–26
• S Boissière, M A Nieper-Wisskirchen, Generating series in the cohomology of Hilbert schemes of points on surfaces, LMS J. Comput. Math. 10 (2007) 254–270
• S Cappell, L Maxim, T Ohmoto, J Schürmann, S Yokura, Characteristic classes of Hilbert schemes of points via symmetric products, Geom. Topol. 17 (2013) 1165–1198
• J Cheah, Cellular decompositions for nested Hilbert schemes of points, Pacific J. Math. 183 (1998) 39–90
• A Douady, Le problème des modules pour les sous-espaces analytiques compacts d'un espace analytique donné, Ann. Inst. Fourier $($Grenoble$)$ 16 (1966) 1–95
• G Ellingsrud, L G öttsche, M Lehn, On the cobordism class of the Hilbert scheme of a surface, J. Algebraic Geom. 10 (2001) 81–100
• J Fogarty, Algebraic families on an algebraic surface, Amer. J. Math 90 (1968) 511–521
• W Fulton, Intersection theory, 2nd edition, Ergeb. Math. Grenzgeb. 2, Springer (1998)
• G González-Sprinberg, L'obstruction locale d'Euler et le théorème de MacPherson, from “Caractéristique d'Euler-Poincaré” (J-L Verdier, editor), Astérisque 83, Soc. Math. France, Paris (1981) 7–32
• L G öttsche, A conjectural generating function for numbers of curves on surfaces, Comm. Math. Phys. 196 (1998) 523–533
• G-M Greuel, C Lossen, E Shustin, Introduction to singularities and deformations, Springer (2007)
• I Grojnowski, Instantons and affine algebras, I: The Hilbert scheme and vertex operators, preprint (1995)
• A Grothendieck, Techniques de construction et théorèmes d'existence en géométrie algébrique, IV: Les schémas de Hilbert, from “Séminaire Bourbaki $1960/1961$ (Exposé 221)”, W A Benjamin, New York (1966) Reprinted as pages 249–276 in Séminaire Bourbaki 6, Soc. Math. France, Paris, 1995
• A Hatcher, Algebraic topology, Cambridge University Press (2002)
• Y Hinohara, K Takahashi, H Terakawa, On tensor products of $k$–very ample line bundles, Proc. Amer. Math. Soc. 133 (2005) 687–692
• M È Kazaryan, Multisingularities, cobordisms, and enumerative geometry, Uspekhi Mat. Nauk 58 (2003) 29–88 In Russian; translated in Russian Math. Surveys 58 (2003) 665–724
• S L Kleiman, The transversality of a general translate, Compositio Math. 28 (1974) 287–297
• S Kleiman, R Piene, Enumerating singular curves on surfaces, from “Algebraic geometry: Hirzebruch 70” (P Pragacz, M Szurek, J Wiśniewski, editors), Contemp. Math. 241, Amer. Math. Soc., Providence, RI (1999) 209–238
• S Kleiman, R Piene, Enriques diagrams, arbitrarily near points, and Hilbert schemes, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl. 22 (2011) 411–451
• J Kollár, Rational curves on algebraic varieties, Ergeb. Math. Grenzgeb. 32, Springer, Berlin (1996)
• M Kool, V Shende, R P Thomas, A short proof of the Göttsche conjecture, Geom. Topol. 15 (2011) 397–406
• M Lehn, Chern classes of tautological sheaves on Hilbert schemes of points on surfaces, Invent. Math. 136 (1999) 157–207
• J Li, Zero dimensional Donaldson–Thomas invariants of threefolds, Geom. Topol. 10 (2006) 2117–2171
• J Li, Y-j Tzeng, Universal polynomials for singular curves on surfaces, Compos. Math. 150 (2014) 1169–1182
• A-K Liu, Family blowup formula, admissible graphs and the enumeration of singular curves, I, J. Differential Geom. 56 (2000) 381–579
• R D MacPherson, Chern classes for singular algebraic varieties, Ann. of Math. 100 (1974) 423–432
• J I Magnússon, A global morphism from the Douady space to the cycle space, Math. Scand. 101 (2007) 19–28
• D Maulik, Stable pairs and the HOMFLY polynomial, preprint (2012)
• H Nakajima, Heisenberg algebra and Hilbert schemes of points on projective surfaces, Ann. of Math. 145 (1997) 379–388
• M A Nieper-Wisskirchen, Characteristic classes of the Hilbert schemes of points on non-compact simply-connected surfaces, JP J. Geom. Topol. 8 (2008) 7–21
• A Oblomkov, V Shende, The Hilbert scheme of a plane curve singularity and the HOMFLY polynomial of its link, Duke Math. J. 161 (2012) 1277–1303
• R Pandharipande, R P Thomas, Stable pairs and BPS invariants, J. Amer. Math. Soc. 23 (2010) 267–297
• A Parusiński, P Pragacz, Chern–Schwartz–MacPherson classes and the Euler characteristic of degeneracy loci and special divisors, J. Amer. Math. Soc. 8 (1995) 793–817
• R P Stanley, Enumerative combinatorics, 1, 2nd edition, Cambridge Studies in Advanced Mathematics 49, Cambridge University Press (2012)
• Y-J Tzeng, A proof of the Göttsche–Yau–Zaslow formula, J. Differential Geom. 90 (2012) 439–472
• T Yasuda, Flag higher Nash blowups, Comm. Algebra 37 (2009) 1001–1015