Big and Nef Tautological Vector Bundles over the Hilbert Scheme of Points

Dragos Oprea

Symmetry, integrability and geometry: methods and applications, Tome 18 (2022)

Résumé

We study tautological vector bundles over the Hilbert scheme of points on surfaces. For each $K$-trivial surface, we write down a simple criterion ensuring that the tautological bundles are big and nef, and illustrate it by examples. In the $K3$ case, we extend recent constructions and results of Bini, Boissière and Flamini from the Hilbert scheme of $2$ and $3$ points to an arbitrary number of points. Among the $K$-trivial surfaces, the case of Enriques surfaces is the most involved. Our techniques apply to other smooth projective surfaces, including blowups of $K3$s and minimal surfaces of general type, as well as to the punctual Quot schemes of curves.

Détail

Keywords: Hilbert scheme, Quot scheme, tautological bundles.

Bibliographie

[1] Aprodu M., Farkas G., Ortega A., “Minimal resolutions, Chow forms and Ulrich bundles on $K3$ surfaces”, J. Reine Angew. Math., 730 (2017), 225–249, arXiv: 1212.6248 | DOI | MR

[2] Arbesfeld N., “K-theoretic Donaldson–Thomas theory and the Hilbert scheme of points on a surface”, Algebr. Geom., 8 (2021), 587–625, arXiv: 1905.04567 | DOI | MR

[3] Arcara D., Bertram A., Coskun I., Huizenga J., “The minimal model program for the Hilbert scheme of points on $\mathbb{P}^2$ and Bridgeland stability”, Adv. Math., 235 (2013), 580–626, arXiv: 1203.0316 | DOI | MR

[4] Bayer A., Macrì E., “Projectivity and birational geometry of Bridgeland moduli spaces”, J. Amer. Math. Soc., 27 (2014), 707–752, arXiv: 1203.4613 | DOI | MR

[5] Beltrametti M., Sommese A.J., “Zero cycles and $k$th order embeddings of smooth projective surfaces (with an appendix by Lothar Göttsche)”, Problems in the Theory of Surfaces and their Classification (Cortona, 1988), Sympos. Math., 32, Academic Press, London, 1991, 33–48 | MR

[6] Bertram A., Coskun I., “The birational geometry of the Hilbert scheme of points on surfaces”, Birational Geometry, Rational Curves, and Arithmetic, Simons Symp., Springer, Cham, 2013, 15–55 | DOI | MR

[7] Bini G., Boissière S., Flamini F., Some families of big and stable bundles on $K3$ surfaces and on their Hilbert schemes of points, arXiv: 2109.01598 | MR

[8] Birkenhake C., Lange H., Complex tori, Progress in Mathematics, 177, Birkhäuser Boston, Inc., Boston, MA, 1999 | DOI | MR

[9] Bolognese B., Huizenga J., Lin Y., Riedl E., Schmidt B., Woolf M., Zhao X., “Nef cones of Hilbert schemes of points on surfaces”, Algebra Number Theory, 10 (2016), 907–930, arXiv: 1509.04722 | DOI | MR

[10] Catanese F., Göttsche L., “$d$-very-ample line bundles and embeddings of Hilbert schemes of $0$-cycles”, Manuscripta Math., 68 (1990), 337–341 | DOI | MR

[11] Coskun I., Nuer H., Yoshioka K., The cohomology of the general stable sheaf on a $K3$ surface, arXiv: 2106.13047

[12] Danila G., “Sections de la puissance tensorielle du fibré tautologique sur le schéma de Hilbert des points d'une surface”, Bull. Lond. Math. Soc., 39 (2007), 311–316, arXiv: math.AG/0104005 | DOI | MR

[13] Demailly J.-P., “Vanishing theorems for tensor powers of an ample vector bundle”, Invent. Math., 91 (1988), 203–220 | DOI | MR

[14] Ellingsrud G., Göttsche L., Lehn M., “On the cobordism class of the Hilbert scheme of a surface”, J. Algebraic Geom., 10 (2001), 81–100, arXiv: math.AG/9904095 | MR

[15] Gangopadhyay C., Sebastian R., “Nef cones of some Quot schemes on a smooth projective curve”, C. R. Math. Acad. Sci. Paris, 359 (2021), 999–1022, arXiv: 2006.16666 | DOI | MR

[16] Göttsche L., “Blowup formulas, strange duality and generating functions for Segre and Verlinde numbers of surfaces”, Intercontinental Moduli and Algebraic Geometry Zoominar (January 17, 2022)

[17] Göttsche L., Kool M., “Virtual Segre and Verlinde numbers of projective surfaces”, Proc. Lond. Math. Soc. (to appear) | MR

[18] Knutsen A.L., “A note on Seshadri constants on general $K3$ surfaces”, C. R. Math. Acad. Sci. Paris, 346 (2008), 1079–1081, arXiv: 0704.2377 | DOI | MR

[19] Kopper J., “The nef cone of the Hilbert scheme of points on rational elliptic surfaces and the cone conjecture”, Canad. Math. Bull., 64 (2021), 216–227, arXiv: 1905.04570 | DOI | MR

[20] Lazarsfeld R., Positivity in algebraic geometry. {I} Classical setting: line bundles and linear series, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 48, Springer-Verlag, Berlin, 2004 | DOI | MR

[21] Lazarsfeld R., Positivity in algebraic geometry. II. Positivity for vector bundles, and multiplier ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 49, Springer-Verlag, Berlin, 2004 | DOI | MR

[22] Lehn M., “Chern classes of tautological sheaves on Hilbert schemes of points on surfaces”, Invent. Math., 136 (1999), 157–207, arXiv: math.AG/9803091 | DOI | MR

[23] Li W.-P., Qin Z., Zhang Q., “Curves in the Hilbert schemes of points on surfaces”, Vector Bundles and Representation Theory (Columbia, MO, 2002), Contemp. Math., 322, Amer. Math. Soc., Providence, RI, 2003, 89–96 | DOI | MR

[24] Maciocia A., Meachan C., “Rank 1 Bridgeland stable moduli spaces on a principally polarized abelian surface”, Int. Math. Res. Not., 2013 (2013), 2054–2077, arXiv: 1107.5304 | DOI | MR

[25] Marian A., Oprea D., Pandharipande R., “Segre classes and Hilbert schemes of points”, Ann. Sci. Éc. Norm. Supér. (4), 50 (2017), 239–267, arXiv: 1507.00688 | DOI | MR

[26] Marian A., Oprea D., Pandharipande R., “The combinatorics of Lehn's conjecture”, J. Math. Soc. Japan, 71 (2019), 299–308, arXiv: 1708.08129 | DOI | MR

[27] Marian A., Oprea D., Pandharipande R., “Higher rank Segre integrals over the Hilbert scheme of points”, J. Eur. Math. Soc. (JEMS), 24 (2022), 2979–3015, arXiv: 1712.02382 | DOI | MR

[28] Mella M., Palleschi M., “The $k$-very ampleness on an elliptic quasi bundle”, Abh. Math. Sem. Univ. Hamburg, 63 (1993), 215–226 | DOI | MR

[29] Mukai S., “Semi-homogeneous vector bundles on an Abelian variety”, J. Math. Kyoto Univ., 18 (1978), 239–272 | DOI | MR

[30] Nuer H., “Projectivity and birational geometry of Bridgeland moduli spaces on an Enriques surface”, Proc. Lond. Math. Soc., 113 (2016), 345–386, arXiv: 1406.0908 | DOI | MR

[31] Oberdieck G., Universality of descendent integrals over moduli spaces of stable sheaves on $K3$ surfaces, arXiv: 2201.03833 | MR

[32] Oprea D., “The Verlinde bundles and the semihomogeneous Wirtinger duality”, J. Reine Angew. Math., 654 (2011), 181–217, arXiv: 0812.4962 | DOI | MR

[33] Oprea D., Pandharipande R., “Quot schemes of curves and surfaces: virtual classes, integrals, Euler characteristics”, Geom. Topol., 25 (2021), 3425–3505, arXiv: 1903.08787 | DOI | MR

[34] Qin Z., Tu Y., “The nef cones of and minimal-degree curves in the Hilbert schemes of points on certain surfaces”, Pacific J. Math., 284 (2016), 439–453 | DOI | MR

[35] Scala L., Higher symmetric powers of tautological bundles on Hilbert schemes of points on a surface, arXiv: 1502.07595

[36] Strømme S.A., “On parametrized rational curves in Grassmann varieties”, Space Curves (Rocca di Papa, 1985), Lecture Notes in Math., 1266, Springer, Berlin, 1987, 251–272 | DOI | MR

[37] Szemberg T., “On positivity of line bundles on Enriques surfaces”, Trans. Amer. Math. Soc., 353 (2001), 4963–4972 | DOI | MR

[38] Voisin C., “Segre classes of tautological bundles on Hilbert schemes of surfaces”, Algebr. Geom., 6 (2019), 186–195, arXiv: 1708.06325 | DOI | MR

[39] Yanagida S., Yoshioka K., “Bridgeland's stabilities on abelian surfaces”, Math. Z., 276 (2014), 571–610, arXiv: 1203.0884 | DOI | MR

[40] Yang X., “Big vector bundles and complex manifolds with semi-positive tangent bundles”, Math. Ann., 367 (2017), 251–282, arXiv: 1412.5156 | DOI | MR

[41] Zhou J., K-theory of Hilbert schemes as a formal quantum field theory, arXiv: 1803.06080