Hostname: page-component-745bb68f8f-grxwn Total loading time: 0 Render date: 2025-01-09T23:12:43.122Z Has data issue: false hasContentIssue false
  • English
  • Français

K-MODULI OF CURVES ON A QUADRIC SURFACE AND K3 SURFACES

Part of: Families, fibrations Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  16 September 2021

Kenneth Ascher Open the ORCID record for Kenneth Ascher [Opens in a new window] ,
Kristin Devleming  and
Yuchen Liu Open the ORCID record for Yuchen Liu [Opens in a new window]
Kenneth Ascher*
Affiliation:
Department of Mathematics, University of California – Irvine, 340 Rowland Hall, Irvine, CA 92697, USA
Kristin Devleming
Affiliation:
Department of Mathematics, University of California – San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA ([email protected])
Yuchen Liu
Affiliation:
Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208, USA ([email protected])
*
[email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that the K-moduli spaces of log Fano pairs $\left(\mathbb {P}^1\times \mathbb {P}^1, cC\right)$ , where C is a $(4,4)$ curve and their wall crossings coincide with the VGIT quotients of $(2,4)$ , complete intersection curves in $\mathbb {P}^3$ . This, together with recent results by Laza and O’Grady, implies that these K-moduli spaces form a natural interpolation between the GIT moduli space of $(4,4)$ curves on $\mathbb {P}^1\times \mathbb {P}^1$ and the Baily–Borel compactification of moduli of quartic hyperelliptic K3 surfaces.

Keywords

K-modulimoduliK3 surfacesvariation of GIT

MSC classification

Primary: 14J10: Families, moduli, classification 14J28: $K3$ surfaces and Enriques surfaces
Secondary: 14D23: Stacks and moduli problems
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 22 , Issue 3 , May 2023 , pp. 1251 - 1291
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Akhtar, M., Coates, T., Corti, A., Heuberger, L., Kasprzyk, A., Oneto, A., Petracci, A., Prince, T. and Tveiten, K., Mirror symmetry and the classification of orbifold del Pezzo surfaces, Proc. Amer. Math. Soc. 144(2) (2016), 513–527.CrossRefGoogle Scholar
Alexeev, V. and Nikulin, V. V., Del Pezzo and $\mathrm{K}3$ surfaces, MSJ Memoirs, 15 (Mathematical Society of Japan, Tokyo, 2006).Google Scholar
Alper, J., Good moduli spaces for Artin stacks, Ann. Inst. Fourier (Grenoble) 63(6) (2013), 2349–2402.CrossRefGoogle Scholar
Alper, J., Blum, H., Halpern-Leistner, D. and Xu, C., Reductivity of the automorphism group of K-polystable Fano varieties, Invent. Math. 222(3) (2020), 995–1032.CrossRefGoogle Scholar
Alper, J., Fedorchuk, M. and Smyth, D. I., Second flip in the Hassett-Keel program: Existence of good moduli spaces, Compos. Math. 153(8) (2017), 1584–1609.CrossRefGoogle Scholar
Ascher, K., DeVleming, K. and Liu, Y., ‘Wall crossing for K-moduli spaces of plane curves’, Preprint, 2019, https://arxiv.org/abs/1909.04576.Google Scholar
Ascher, K., DeVleming, K. and Liu, Y., ‘K-stability and birational models of moduli of quartic K3 surfaces’, 2021, in preparation. https://arxiv.org/abs/2108.06848.CrossRefGoogle Scholar
Benoist, O., Quelques espaces de modules d’intersections complètes lisses qui sont quasi-projectifs, J. Eur. Math. Soc. (JEMS) 16(8) (2014), 1749–1774.CrossRefGoogle Scholar
Blum, H., Existence of valuations with smallest normalized volume, Compos. Math. 154(4) (2018), 820–849.CrossRefGoogle Scholar
Blum, H. and Xu, C., Uniqueness of $\mathsf{K}$ -polystable degenerations of Fano varieties, Ann. of Math. (2) 190(2) (2019), 609–656.CrossRefGoogle Scholar
Blum, H., Halpern-Leistner, D., Liu, Y. and Xu, C., ‘On properness of K-moduli spaces and optimal degenerations of Fano varieties’, Preprint, 2020, https://arxiv.org/abs/2011.01895.CrossRefGoogle Scholar
Blum, H., Liu, Y. and Xu, C., ‘Openness of K-semistability for Fano varieties’, Preprint, 2019, https://arxiv.org/abs/1907.02408.Google Scholar
Boucksom, S., de Fernex, T., Favre, C. and Urbinati, S., Valuation spaces and multiplier ideals on singular varieties, in Recent Advances in Algebraic Geometry, London Mathematical Society Lecture Note Series, 417, pp. 29–51 (Cambridge University Press, Cambridge, UK, 2015).CrossRefGoogle Scholar
Boucksom, S., Hisamoto, T. and Jonsson, M., Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs, Ann. Inst. Fourier (Grenoble) 67(2) (2017), 743–841.CrossRefGoogle Scholar
Xu, C., A minimizing valuation is quasi-monomial, Ann. of Math. (2) 191(3) (2020), 1003–1030.Google Scholar
Casalaina-Martin, S., Jensen, D. and Laza, R., Log canonical models and variation of GIT for genus 4 canonical curves, J. Algebraic Geom. 23(4) (2014), 727–764.Google Scholar
Codogni, G. and Patakfalvi, Z., Positivity of the CM line bundle for families of K-stable klt Fano varieties, Invent. Math. 223(3) (2021), 811–894.CrossRefGoogle Scholar
Dervan, R., On K-stability of finite covers, Bull. Lond. Math. Soc. 48(4) (2016), 717–728.CrossRefGoogle Scholar
Dolgachev, I. V. and Hu, Y, Variation of geometric invariant theory quotients, Publ. Math. Inst. Hautes Études Sci. 87(1998), 5–56. With an appendix by Nicolas Ressayre.CrossRefGoogle Scholar
Donaldson, S. K., Scalar curvature and stability of toric varieties, J. Differential Geom. 62(2) (2002), 289–349.CrossRefGoogle Scholar
Ein, L., Lazarsfeld, R. and Smith, K. E., Uniform approximation of Abhyankar valuation ideals in smooth function fields, Amer. J. Math. 125(2) (2003), 409–440.CrossRefGoogle Scholar
Fine, J. and Ross, J., A note on positivity of the CM line bundle, Int. Math. Res. Not. IMRN (2006), 95875.CrossRefGoogle Scholar
Fujiki, A. and Schumacher, G., The moduli space of extremal compact Kähler manifolds and generalized Weil-Petersson metrics, Publ. Res. Inst. Math. Sci. 26(1) (1990), 101–183.CrossRefGoogle Scholar
Fujita, K., ‘K-stability of log Fano hyperplane arrangements’, Preprint, 2017, https://arxiv.org/abs/1709.08213.Google Scholar
Fujita, K., Optimal bounds for the volumes of Kähler-Einstein Fano manifolds’, Amer. J. Math. 140(2) (2018), 391–414.CrossRefGoogle Scholar
Fujita, K., A valuative criterion for uniform K-stability of $\mathbb{Q}$ -Fano varieties, J. Reine Angew. Math. 751 (2019), 309–338.CrossRefGoogle Scholar
Fujita, K., On K-polystability for log del Pezzo pairs of Maeda type, Acta Math. Vietnam. 45(4) (2020), 943–965.CrossRefGoogle Scholar
Gallardo, P., Martinez-Garcia, J. and Spotti, C., Applications of the moduli continuity method to log K-stable pairs, J. Lond. Math. Soc. (2) 103(2) (2021), 729–759.CrossRefGoogle Scholar
Hacking, P., Compact moduli of plane curves, Duke Math. J. 124(2) (2004), 213–257.CrossRefGoogle Scholar
Hacking, P. and Prokhorov, Y., ‘Degenerations of del Pezzo surfaces I’, Preprint, 2005, https://arxiv.org/abs/math/0509529.Google Scholar
Hacking, P. and Prokhorov, Y., Smoothable del Pezzo surfaces with quotient singularities, Compos. Math. 146(1) (2010), 169–192.CrossRefGoogle Scholar
Hacon, C. D., McKernan, J. and Xu, C., ACC for log canonical thresholds, Ann. of Math. (2) 180(2) (2014), 523–571.CrossRefGoogle Scholar
Jiang, C., Boundedness of $\mathbb{Q}$ -Fano varieties with degrees and alpha-invariants bounded from below, Ann. Sci. Éc. Norm. Supér. (4) 53(5) (2020), 1235–1248.CrossRefGoogle Scholar
Jonsson, M. and Mustaţă, M., Valuations and asymptotic invariants for sequences of ideals, Ann. Inst. Fourier (Grenoble) 62(6) (2012), 2145–2209 (2013).CrossRefGoogle Scholar
Kempf, G. R., Instability in invariant theory, Ann. of Math. (2) 108(2) (1978), 299–316.CrossRefGoogle Scholar
Knudsen, F. F. and Mumford, D., The projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div”, Math. Scand. 39(1) (1976), 19–55.CrossRefGoogle Scholar
Kollár, J., Families of Varieties of General Type, 2017, https://web.math.princeton.edu/~kollar/book/modbook20170720-hyper.pdf.Google Scholar
Kollár, J., ‘Mumford divisors’, Preprint, 2018, https://arxiv.org/abs/1803.07596.Google Scholar
Kollár, J., ‘Families of divisors’, Preprint, 2019, https://arxiv.org/abs/1910.00937.Google Scholar
Kollár, J. and Mori, S., Birational Geometry of Algebraic Varieties , Cambridge Tracts in Mathematics, vol. 134 (Cambridge University Press, Cambridge, UK, 1998). With the collaboration of C. H. Clemens and A. Corti. Translated from the 1998 Japanese original.Google Scholar
Laza, R., GIT and moduli with a twist, in Handbook of Moduli, Vol. II, Advanced Lectures in Mathematics (ALM), 25, pp. 259–297 (International Press, Somerville, MA, 2013).Google Scholar
Laza, R. and O’Grady, K., Birational geometry of the moduli space of quartic $K3$ surfaces, Compos. Math. 155(9) (2019), 1655–1710.CrossRefGoogle Scholar
Laza, R. and O’Grady, K., GIT versus Baily-Borel compactification for $K3$ ’s which are double covers of ${\mathbb{P}}^1\times {\mathbb{P}}^1$ , Adv. Math. 383 (2021), 107680.Google Scholar
Laza, R. and O’Grady, K. G., GIT versus Baily-Borel compactification for quartic $K3$ surfaces, in Geometry of Moduli, Abel Symposia, 14, pp. 217–283 (Springer, Cham, Switzerland, 2018).CrossRefGoogle Scholar
Laza, R. and Zhang, Z., Classical period domains, in Recent Advances in Hodge Theory, London Mathematical Society Lecture Note Series, 427, pp. 3–44 (Cambridge University Press, Cambridge, UK, 2016).CrossRefGoogle Scholar
Li, C., Remarks on logarithmic K-stability, Commun. Contemp. Math. 17(2) (2015), 1450020.CrossRefGoogle Scholar
Li, C., K-semistability is equivariant volume minimization, Duke Math. J. 166(16) (2017), 3147–3218.CrossRefGoogle Scholar
Li, C., Minimizing normalized volumes of valuations, Math. Z. 289(1-2) (2018), 491–513.CrossRefGoogle Scholar
Li, C. and Xu, C., Special test configuration and K-stability of Fano varieties, Ann. of Math. (2) 180(1) (2014), 197–232.CrossRefGoogle Scholar
Li, C. and Xu, C., Stability of valuations and Kollár components, J. Eur. Math. Soc. (JEMS) 22(8) (2020), 2573–2627.CrossRefGoogle Scholar
Li, C. and Liu, Y., Kähler-Einstein metrics and volume minimization, Adv. Math. 341 (2019), 440–492.CrossRefGoogle Scholar
Li, C., Liu, Y. and Xu, C., A guided tour to normalized volume, in Geometric Analysis, Progress in Mathematics, 333, pp 167–219 (Birkhäuser/Springer, Cham, Switzerland, 2020).CrossRefGoogle Scholar
Li, C., Wang, X. and Xu, C., Algebraicity of the metric tangent cones and equivariant K-stability, J. Amer. Math. Soc. 34(4) (2021), 1175–1214.CrossRefGoogle Scholar
Liu, Y., The volume of singular Kähler-Einstein Fano varieties, Compos. Math. 154(6) (2018), 1131–1158.CrossRefGoogle Scholar
Liu, Y., ‘K-stability of cubic fourfolds’, Preprint, 2020, https://arxiv.org/abs/2007.14320.Google Scholar
Liu, Y. and Xu, C., K-stability of cubic threefolds, Duke Math. J. 168(11) (2019), 2029–2073.CrossRefGoogle Scholar
Liu, Y., Xu, C. and Zhuang, Z., ‘Finite generation for valuations computing stability thresholds and applications to K-stability’, Preprint, 2021, https://arxiv.org/abs/2102.09405.Google Scholar
Looijenga, E., New compactifications of locally symmetric varieties, in Proceedings of the 1984 Vancouver Conference in Algebraic Geometry, CMS Conference Proceedings, 6, pp. 341–364 (American Mathematical Society, Providence, RI, 1986).Google Scholar
Looijenga, E., Compactifications defined by arrangements. I. The ball quotient case, Duke Math. J. 118(1) (2003), 151–187.CrossRefGoogle Scholar
Mabuchi, T. and Mukai, S., Stability and Einstein-Kähler metric of a quartic del Pezzo surface, in Einstein Metrics and Yang-Mills Connections (Sanda, 1990), Lecture Notes in Pure and Applied Mathematics, 145, pp. 133–160 (Dekker, New York, 1993).Google Scholar
Mumford, D., Fogarty, J. and Kirwan, F., Geometric Invariant Theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2), 34 (Springer-Verlag, Berlin, 1994).CrossRefGoogle Scholar
Nakayama, N., Classification of log del Pezzo surfaces of index two, J. Math. Sci. Univ. Tokyo 14(3) (2007), 293–498.Google Scholar
Odaka, Y., The Calabi conjecture and K-stability, Int. Math. Res. Not. IMRN 2012(10) (2012), 2272–2288.Google Scholar
Odaka, Y., A generalization of the Ross-Thomas slope theory, Osaka J. Math. 50(1) (2013), 171–185.Google Scholar
Odaka, Y., Spotti, C. and Sun, S., Compact moduli spaces of del Pezzo surfaces and Kähler-Einstein metrics, J. Differential Geom. 102(1) (2016), 127–172.CrossRefGoogle Scholar
Odaka, Y. and Sun, S., Testing log K-stability by blowing up formalism, Ann. Fac. Sci. Toulouse Math. (6) 24(3) (2015), 505–522.CrossRefGoogle Scholar
Paul, S. T. and Tian, G., ‘CM stability and the generalized Futaki invariant I’, Preprint, 2006, https://arxiv.org/abs/math/0605278.Google Scholar
Paul, S. T. and Tian, G., CM stability and the generalized Futaki invariant II, Astérisque, No 328 (2009), p. 339-354.Google Scholar
Shah, J., A complete moduli space for $K3$ surfaces of degree $2$ , Ann. of Math. (2) 112(3) (1980), 485–510.CrossRefGoogle Scholar
Skoda, H., Sous-ensembles analytiques d’ordre fini ou infini dans ${C}^n$ , Bull. Soc. Math. France 100 (1972), 353–408.CrossRefGoogle Scholar
Spotti, C. and Sun, S., Explicit Gromov-Hausdorff compactifications of moduli spaces of Kähler-Einstein Fano manifolds, Pure Appl. Math. Q. 13(3) (2017), 477–515.CrossRefGoogle Scholar
Thaddeus, M., Geometric invariant theory and flips, J. Amer. Math. Soc. 9(3) (1996), 691–723.CrossRefGoogle Scholar
The Stacks Project Authors, ‘Stacks project’, 2018, https://stacks.math.columbia.edu.Google Scholar
Tian, G., Kähler-Einstein metrics with positive scalar curvature, Invent. Math. 130(1) (1997), 1–37.CrossRefGoogle Scholar
Wang, X., Height and GIT weight, Math. Res. Lett. 19(4) (2012), 909–926.CrossRefGoogle Scholar
Xu, C. and Zhuang, Z., On positivity of the CM line bundle on K-moduli spaces, Ann. of Math. (2) 192(3) (2020), 1005–1068.CrossRefGoogle Scholar
Xu, C. and Zhuang, Z., Uniqueness of the minimizer of the normalized volume function, Camb. J. Math., 9(1) (2021), 149–176.CrossRefGoogle Scholar
Zhuang, Z., Product theorem for K-stability, Adv. Math. 371 (2020), 107250.CrossRefGoogle Scholar