The desingularization of the theta divisor of a cubic threefold as a moduli space

Bayer, Arend¹ ; Beentjes, Sjoerd Viktor¹ ; Feyzbakhsh, Soheyla² ; Hein, Georg³ ; Martinelli, Diletta⁴ ; Rezaee, Fatemeh⁵ ; Schmidt, Benjamin⁶

¹ School of Mathematics and Maxwell Institute, University of Edinburgh, Edinburgh, United Kingdom
² Department of Mathematics, Imperial College London, London, United Kingdom
³ Fakultät für Mathematik, Universität Duisburg-Essen, Essen, Germany
⁴ Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Amsterdam, Netherlands
⁵ Centre for Mathematical Sciences, University of Cambridge, Cambridge, United Kingdom, ETH Zürich, Zürich, Switzerland
⁶ Institut für Algebraische Geometrie, Gottfried Wilhelm Leibniz Universität Hannover, Hannover, Germany

Geometry & topology, Tome 28 (2024) no. 1, pp. 127-160.

Résumé

We show that the moduli space ${\overline{M}}_X(v)$ of Gieseker stable sheaves on a smooth cubic threefold $X$ with Chern character $v=\left(3,-H,-\frac{1}{2}{H}^2,\frac{1}{6}{H}^3\right)$ is smooth and of dimension four. Moreover, the Abel–Jacobi map to the intermediate Jacobian of $X$ maps it birationally onto the theta divisor $\Theta$, contracting only a copy of $X\subset {\overline{M}}_X(v)$ to the singular point $0\in \Theta$.

We use this result to give a new proof of a categorical version of the Torelli theorem for cubic threefolds, which says that $X$ can be recovered from its Kuznetsov component $\mathrm{Ku}(X)\subset D^b(X)$. Similarly, this leads to a new proof of the description of the singularity of the theta divisor, and thus of the classical Torelli theorem for cubic threefolds, ie that $X$ can be recovered from its intermediate Jacobian.

Détail

DOI : 10.2140/gt.2024.28.127

Keywords: cubic threefolds, derived categories, stability conditions

Bibliographie

[1] M Altavilla, M Petković, F Rota, Moduli spaces on the Kuznetsov component of Fano threefolds of index 2, Épijournal Géom. Algébrique 6 (2022) 13 | DOI

[2] D Arcara, A Bertram, Bridgeland-stable moduli spaces for K–trivial surfaces, J. Eur. Math. Soc. 15 (2013) 1 | DOI

[3] M Artebani, R Kloosterman, M Pacini, A new model for the theta divisor of the cubic threefold, Matematiche (Catania) 58 (2003) 201

[4] M Artin, Algebraization of formal moduli, II : Existence of modifications, Ann. of Math. 91 (1970) 88 | DOI

[5] A Bayer, M Lahoz, E Macrì, P Stellari, Stability conditions on Kuznetsov components, Ann. Sci. Éc. Norm. Supér. 56 (2023) 517 | DOI

[6] A Bayer, E Macrì, P Stellari, The space of stability conditions on abelian threefolds, and on some Calabi–Yau threefolds, Invent. Math. 206 (2016) 869 | DOI

[7] A Bayer, E Macrì, Y Toda, Bridgeland stability conditions on threefolds, I : Bogomolov–Gieseker type inequalities, J. Algebraic Geom. 23 (2014) 117 | DOI

[8] A Beauville, Les singularités du diviseur Θ de la Jacobienne intermédiaire de l’hypersurface cubique dans P4, from: "Algebraic threefolds" (editor A Conte), Lecture Notes in Math. 947, Springer (1982) 190 | DOI

[9] A Beauville, Vector bundles on the cubic threefold, from: "Symposium in honor of C H Clemens" (editors A Bertram, J A Carlson, H Kley), Contemp. Math. 312, Amer. Math. Soc. (2002) 71 | DOI

[10] P Belmans, A J De Jong, Others, The Stacks project, electronic reference (2005–)

[11] M Bernardara, E Macrì, S Mehrotra, P Stellari, A categorical invariant for cubic threefolds, Adv. Math. 229 (2012) 770 | DOI

[12] T Bridgeland, Stability conditions on triangulated categories, Ann. of Math. 166 (2007) 317 | DOI

[13] T Bridgeland, Stability conditions on K3 surfaces, Duke Math. J. 141 (2008) 241 | DOI

[14] C H Clemens, P A Griffiths, The intermediate Jacobian of the cubic threefold, Ann. of Math. 95 (1972) 281 | DOI

[15] I Coskun, J Huizenga, The ample cone of moduli spaces of sheaves on the plane, Algebr. Geom. 3 (2016) 106 | DOI

[16] L Ein, R Lazarsfeld, Singularities of theta divisors and the birational geometry of irregular varieties, J. Amer. Math. Soc. 10 (1997) 243 | DOI

[17] S Feyzbakhsh, L Pertusi, Serre-invariant stability conditions and Ulrich bundles on cubic threefolds, Épijournal Géom. Algébrique 7 (2023) 1 | DOI

[18] D Gieseker, On the moduli of vector bundles on an algebraic surface, Ann. of Math. 106 (1977) 45 | DOI

[19] P A Griffiths, Periods of integrals on algebraic manifolds, II : Local study of the period mapping, Amer. J. Math. 90 (1968) 805 | DOI

[20] D Happel, I Reiten, S O Smalø, Tilting in abelian categories and quasitilted algebras, 575, Amer. Math. Soc. (1996) | DOI

[21] R Hartshorne, Algebraic geometry, 52, Springer (1977) | DOI

[22] R Hartshorne, Generalized divisors on Gorenstein schemes, –Theory 8 (1994) 287 | DOI

[23] D Huybrechts, M Lehn, The geometry of moduli spaces of sheaves, Cambridge Univ. Press (2010) | DOI

[24] A Iliev, Minimal sections of conic bundles, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. 2 (1999) 401

[25] A G Kuznetsov, Derived category of a cubic threefold and the variety V 14, Tr. Mat. Inst. Steklova 246 (2004) 183

[26] C Li, Stability conditions on Fano threefolds of Picard number 1, J. Eur. Math. Soc. 21 (2019) 709 | DOI

[27] J Lo, Y More, Some examples of tilt-stable objects on threefolds, Comm. Algebra 44 (2016) 1280 | DOI

[28] A Maciocia, Computing the walls associated to Bridgeland stability conditions on projective surfaces, Asian J. Math. 18 (2014) 263 | DOI

[29] E Macrì, B Schmidt, Derived categories and the genus of space curves, Algebr. Geom. 7 (2020) 153 | DOI

[30] M Maruyama, Moduli of stable sheaves. I, J. Math. Kyoto Univ. 17 (1977) 91 | DOI

[31] M Maruyama, Moduli of stable sheaves. II, J. Math. Kyoto Univ. 18 (1978) 557 | DOI

[32] D Mumford, Projective invariants of projective structures and applications, from: "Proceedings of the International Congress of Mathematicians" (editor V Stenström), Institut Mittag-Leffler (1963) 526

[33] D Mumford, Prym varieties, I, from: "Contributions to analysis : a collection of papers dedicated to Lipman Bers" (editors L V Ahlfors, I Kra, B Maskit, L Nirenberg), Academic (1974) 325

[34] A Perry, The integral Hodge conjecture for two-dimensional Calabi–Yau categories, Compos. Math. 158 (2022) 287 | DOI

[35] L Pertusi, S Yang, Some remarks on Fano threefolds of index two and stability conditions, Int. Math. Res. Not. 2022 (2022) 12940 | DOI

[36] K Schwede, Generalized divisors and reflexive sheaves, lecture notes (2007)

[37] C T Simpson, Moduli of representations of the fundamental group of a smooth projective variety, I, Inst. Hautes Études Sci. Publ. Math. 79 (1994) 47

[38] F Takemoto, Stable vector bundles on algebraic surfaces, Nagoya Math. J. 47 (1972) 29

[39] G E Welters, Abel–Jacobi isogenies for certain types of Fano threefolds, 141, Mathematisch Centrum (1981)

[40] S Zhang, Bridgeland moduli spaces for Gushel–Mukai threefolds and Kuznetsov’s Fano threefold conjecture, preprint (2021)