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Hilbert schemes of lines and conics and automorphism groups of Fano threefolds. (English) Zbl 1406.14031

The paper under review is a thorough survey on Hilbert schemes of lines and conics of Fano threefolds of Picard number one, and on the automorphism groups of such manifolds.
A Fano threefold \(X\) is a smooth variety \(X\) whose anticanonical bundle \(-K_X\) is ample; important invariants of of Fano threefolds are the index \(i_X\), which is the maximum integer such that \(-K_X \sim i_X H\) in the Picard group of \(X\), the degree \(d(X)=H^3\) and the genus, defined as \(g(X)=-\frac{1}{2}K_X^3 +1\), which is an integer greater than or equal to \(2\).
The authors study the Hilbert schemes of lines on del Pezzo threefolds (that is, Fano threefolds of index two) of degree \(d(X) \geq 3\) and the Hilbert scheme of conics on Fano threefolds of index one and genus \(g(X) \geq 7\). In both cases \(H\) is very ample, so lines and conics are lines and conics with respect to the embedding defined by \(|H|\).
Many of the results described in Theorem 1.1.1 where already known, but scattered in the literature over a long timespan; in the reviewer’s opinion it is a commendable effort, which can be very useful for people working in the field, to have them collected and proved all together.
Moreover the proof presented in the paper for the even genus cases emphasizes a correspondence between Fano threefolds of index one and two, which arises at the level of derived categories and has been previously noted by A. G. Kuznetsov [Proc. Steklov Inst. Math. 264, 110–122 (2009; Zbl 1312.14055); translation from Tr. Mat. Inst. Steklova 264, 116–128 (2009)].
The description of the Hilbert schemes is then used to obtain results on the automorphism groups of the variety, since the induced action on the Hilbert schemes is proved to be faithful in all the cases described in Theorem 1.1.1. In particular the authors prove (Theorem 1.1.2) that the automorphism group of a Fano threefold is finite, except for a (short) list of varieties for which the group is explicitly described.

MSC:

14J45 Fano varieties
14J50 Automorphisms of surfaces and higher-dimensional varieties
14J30 \(3\)-folds
14C05 Parametrization (Chow and Hilbert schemes)

Citations:

Zbl 1312.14055
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References:

[1] Adler, A.: On the automorphism group of a certain cubic threefold. Amer. J. Math. 100, 1275-1280 (1978) · Zbl 0405.14019
[2] Altman, A.B., Kleiman, S.L.: Foundations of the theory of Fano schemes. Compositio Math. 34, 3-47 (1977) · Zbl 0414.14024
[3] A. Beauville, Les singularités du diviseur \[\Theta\] Θ de la jacobienne intermédiaire de l’hypersurface cubique dans \[\mathbb{P}^4\] P4, In: Algebraic Threefolds, Varenna, 1981, Lecture Notes in Math., 947, Springer-Verlag, 1982, pp. 190-208 · Zbl 1111.14038
[4] O. Benoist, Séparation et propriété de Deligne-Mumford des champs de modules d’intersections complètes lisses, J. Lond. Math. Soc. (2), 87 (2013), 138-156 · Zbl 1375.14049
[5] Brambilla, M.C., Faenzi, D.: Rank-two stable sheaves with odd determinant on Fano threefolds of genus nine. Math. Z. 275, 185-210 (2013) · Zbl 1286.14059
[6] A. Bondal and D. Orlov, Semiorthogonal decomposition for algebraic varieties, preprint, arXiv:alg-geom/9506012 · Zbl 0822.14021
[7] G. Castelnuovo, Sulle superficie algebriche che ammettono un sistema doppiamente infinito di sezioni piane riduttibili, Rom. Acc. L. Rend. (5), 3 (1894), 22-25 · JFM 25.1228.02
[8] Cheltsov, I., Shramov, C.A.: Cremona Groups and the Icosahedron. Monogr. Res. Notes Math., CRC Press, Boca Raton, FL, (2016) · Zbl 1328.14003
[9] X. Chen, X. Pan and D. Zhang, Automorphism and cohomology II: Complete intersections, preprint, arXiv:1511.07906 · Zbl 0744.14029
[10] M. Cornalba, Una osservazione sulla topologia dei rivestimenti ciclici di varieta algebriche, Boll. Un. Mat. Ital. A (5), 18 (1981), 323-328 · Zbl 0462.14007
[11] Cutkosky, S.D.: On Fano 3-folds. Manuscripta Math. 64, 189-204 (1989) · Zbl 0704.14032
[12] Debarre, O., Kuznetsov, A.: Gushel-Mukai varieties: classification and birationalities. Algebr. Geom. 5, 15-76 (2018) · Zbl 1408.14053
[13] Desale, U.V., Ramanan, S.: Classification of vector bundles of rank 2 on hyperelliptic curves. Invent. Math. 38, 161-185 (1976) · Zbl 0323.14012
[14] Dinew, S., Kapustka, G., Kapustka, M.: Remarks on Mukai threefolds admitting \[\mathbb{C}^*C\]∗ action. Mosc. Math. J. 17, 15-33 (2017) · Zbl 1376.32031
[15] Dolgachev, I.V.: Classical Algebraic Geometry. Cambridge Univ. Press, Cambridge (2012) · Zbl 1252.14001
[16] Dolgachev, I.V., Kanev, V.: Polar covariants of plane cubics and quartics. Adv. Math. 98, 216-301 (1993) · Zbl 0791.14013
[17] A. Fonarev and A. Kuznetsov, Derived categories of curves as components of Fano manifolds, J. Lond. Math. Soc. (2)., 10.1112/jlms.12094 (2017) · Zbl 1468.14037
[18] W. Fulton, Intersection Theory, Ergeb. Math. Grenzgeb. (3), 2, Springer-Verlag, 1984 · Zbl 0541.14005
[19] Furushima, M.: Mukai-Umemura’s example of the Fano threefold with genus \[1212\] as a compactification of \[\mathbf{C}^3\] C3. Nagoya Math. J. 127, 145-165 (1992) · Zbl 0792.14021
[20] Furushima, M., Nakayama, N.: The family of lines on the Fano threefold \[V_5\] V5. Nagoya Math. J. 116, 111-122 (1989) · Zbl 0731.14025
[21] M. Furushima and M. Tada, Nonnormal Del Pezzo surfaces and Fano threefolds of the first kind, J. Reine Angew. Math., 429 (1992), 183-190. · Zbl 0744.14029
[22] D. Fusi, On rational varieties of small degree, preprint, arXiv:1509.06823 · Zbl 1493.14081
[23] González-Aguilera, V., Liendo, A.: Automorphisms of prime order of smooth cubic n-folds. Arch. Math. (Basel) 97, 25-37 (2011) · Zbl 1231.14033
[24] P. Griffiths and J. Harris, Principles of Algebraic Geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978 · Zbl 0408.14001
[25] Grothendieck, A.: Géométrie formelle et géométrie algébrique. Séminaire Bourbaki 5, 193-220 (1958)
[26] A. Grothendieck, Fondements de la géométrie algébrique. [Extraits du Séminaire Bourbaki, 1957-1962], Secrétariat mathématique, Paris, 1962 · Zbl 0239.14001
[27] Gruson, L., Laytimi, F., Nagaraj, D.S.: On prime Fano threefolds of genus 9. Internat. J. Math. 17, 253-261 (2006) · Zbl 1094.14027
[28] R. Hartshorne, Ample Subvarieties of Algebraic Varieties. Notes written in collaboration with C. Musili, Lecture Notes in Math., 156, Springer-Verlag, 1970 · Zbl 0208.48901
[29] Hoppe, H.J.: Generischer Spaltungstyp und zweite Chernklasse stabiler Vektorraumbündel vom Rang 4 auf \[\mathbb{P}_4\] P4. Math. Z. 187, 345-360 (1984) · Zbl 0567.14011
[30] T. Hosoh, Automorphism groups of cubic surfaces, J. Algebra, 192 (1997), 651-677 · Zbl 0910.14021
[31] Hosoh, T.: Equations of specific cubic surfaces and automorphisms. SUT J. Math. 38, 127-134 (2002) · Zbl 1047.14027
[32] J.E. Humphreys, Linear Algebraic Groups, Grad. Texts in Math., 21, Springer-Verlag, 1975 · Zbl 0325.20039
[33] D. Huybrechts and M. Lehn, The Geometry of Moduli Spaces of Sheaves. Second ed., Cambridge Math. Lib., Cambridge Univ. Press, Cambridge, 2010 · Zbl 1206.14027
[34] Iliev, A.: The Fano surface of the Gushel threefold. Compositio Math. 94, 81-107 (1994) · Zbl 0822.14021
[35] Iliev, A.: The \[Sp_3\] Sp3-Grassmannian and duality for prime Fano threefolds of genus 9. Manuscripta Math. 112, 29-53 (2003) · Zbl 1078.14528
[36] Iliev, A., Manivel, L.: Prime Fano threefolds and integrable systems. Math. Ann. 339, 937-955 (2007) · Zbl 1136.14026
[37] A. Iliev and L. Manivel, Fano manifolds of degree ten and EPW sextics, Ann. Sci. Éc. Norm. Supér. (4), 44 (2011), 393-426 · Zbl 1258.14050
[38] V.A. Iskovskikh, Anticanonical models of three-dimensional algebraic varieties, J. Soviet Math., 13 (1980), 745-814 · Zbl 0428.14016
[39] V.A. Iskovskikh, Lectures on Three-Dimensional Algebraic Varieties. Fano Varieties. (Lektsii po trekhmernym algebraicheskim mnogoobraziyam. Mnogoobraziya Fano), Izdatel’stvo Moskovskogo Univ., Moskva, 1988 · Zbl 0698.14041
[40] Iskovskikh, V.A.: Double projection from a line onto Fano 3-folds of the first kind. Mat. Sb. 180, 260-278 (1989)
[41] V.A. Iskovskikh and Yu.G. Prokhorov, Fano Varieties. Algebraic Geometry V, Encyclopaedia Math. Sci., 47, Springer-Verlag, 1999 · Zbl 0323.14012
[42] Kuznetsov, A., Manivel, L., Markushevich, D.: Abel-Jacobi maps for hypersurfaces and noncommutative Calabi-Yau’s. Commun. Contemp. Math. 12, 373-416 (2010) · Zbl 1201.14031
[43] Kollár, J.: Flops. Nagoya Math. J. 113, 15-36 (1989) · Zbl 0645.14004
[44] J. Kollár, Projectivity of complete moduli, J. Differential Geom., 32 (1990), 235-268 · Zbl 0684.14002
[45] J. Kollár, Rational Curves on Algebraic Varieties, Ergeb. Math. Grenzgeb. (3), 32, Springer-Verlag, 1996 · Zbl 0877.14012
[46] J. Kollár and F.-O. Schreyer, Real Fano 3-folds of type \[V_{22}\] V22, In: The Fano Conference, Univ. Torino, Turin, 2004, pp. 515-531 · Zbl 1070.14058
[47] O. Küchle, On Fano 4-fold of index \[11\] and homogeneous vector bundles over Grassmannians, Math. Z., 218 (1995), 563-575 · Zbl 0826.14024
[48] Kuznetsov, A.G.: Exceptional collection of vector bundles on \[V_{22}\] V22 Fano threefolds. Moscow Univ. Math. Bull. 51, 35-37 (1996) · Zbl 0913.14010
[49] A.G. Kuznetsov, Derived categories of cubic and \[V_{14}\] V14 threefolds, Proc. Steklov Inst. Math., 246 (2004), 171-194; In: Algebraic Geometry. Methods, Relations, and Applications. Collected Papers Dedicated to the Memory of Andrei Nikolaevich Tyurin, Maik Nauka/Interperiodica, Moscow, translation from Tr. Mat. Inst. Steklova, 246 (2004), 183-207 · Zbl 0405.14019
[50] Kuznetsov, A.G.: Derived categories of the Fano threefolds \[V_{12}\] V12. Math. Notes 78, 537-550 (2005) · Zbl 1111.14038
[51] A.G. Kuznetsov, Homological projective duality for Grassmannians of lines, preprint, arXiv:math/0610957 · Zbl 0822.14021
[52] Kuznetsov, A.G.: Hyperplane sections and derived categories. Izv. Math. 70, 447-547 (2006) · Zbl 1133.14016
[53] Kuznetsov, A.G.: Derived categories of quadric fibrations and intersections of quadrics. Adv. Math. 218, 1340-1369 (2008) · Zbl 1168.14012
[54] Kuznetsov, A.G.: Derived categories of Fano threefolds. Proc. Steklov Inst. Math. 264, 110-122 (2009) · Zbl 1312.14055
[55] Kuznetsov, A.G.: Instanton bundles on Fano threefolds. Cent. Eur. J. Math. 10, 1198-1231 (2012) · Zbl 1282.14075
[56] A.G. Kuznetsov, Semiorthogonal decompositions in algebraic geometry, In: Proceedings of the International Congress of Mathematicians. Vol. II, Seoul, 2014, pp. 635-660 · Zbl 1373.18009
[57] Kuznetsov, A.G.: On Küchle varieties with Picard number greater than \[11\]. Izv. Math. 79, 698-709 (2015) · Zbl 1342.14087
[58] Kuznetsov, A.G.: Küchle fivefolds of type c5. Math. Z. 284, 1245-1278 (2016) · Zbl 1352.14029
[59] A.G. Kuznetsov, On linear sections of the spinor tenfold, I, preprint, arXiv:1801.00037 · Zbl 1420.14094
[60] A.G. Kuznetsov and Yu.G. Prokhorov, Prime Fano threefolds of genus 12 with a \[\mathbb{G}_{\rm m}\] Gm-action and their automorphisms, preprint, arXiv:1711.08504 · Zbl 1133.14016
[61] D.G. Markushevich, Numerical invariants of families of lines on some Fano varieties, Mat. Sb. (N.S.), 116 (158) (1981), 265-288 · Zbl 0492.14026
[62] M. Maruyama, On boundedness of families of torsion free sheaves, J. Math. Kyoto Univ., 21, (1981) 673-701 · Zbl 0495.14009
[63] H. Matsumura and P. Monsky, On the automorphisms of hypersurfaces, J. Math. Kyoto Univ. 3, (1964) 347-361 · Zbl 0141.37401
[64] Y. Miyaoka and S. Mori, A numerical criterion for uniruledness, Ann. of Math. (2), 124 (1986), 65-69 · Zbl 0606.14030
[65] Mukai, S.: Biregular classification of Fano 3-folds and Fano manifolds of coindex \[33\]. Proc. Nat. Acad. Sci. 86, 3000-3002 (1989) · Zbl 0679.14020
[66] S. Mukai, Fano 3-folds, In: Complex Projective Geometry, Trieste-Bergen, 1989, London Math. Soc. Lecture Note Ser., 179, Cambridge Univ. Press, Cambridge, 1992, pp. 255-263 · Zbl 0774.14037
[67] S. Mukai and H. Umemura, Minimal rational threefolds, In: Algebraic Geometry, Tokyo-Kyoto, 1982, Lecture Notes in Math., 1016, Springer-Verlag, 1983, pp. 490-518 · Zbl 0526.14006
[68] M.S. Narasimhan and S. Ramanan, Moduli of vector bundles on a compact Riemann surface, Ann. Math. (2), 89 (1969), 14-51 · Zbl 0186.54902
[69] K. Oguiso and X. Yu, Automorphism groups of smooth quintic threefolds, preprint, arXiv:1504.05011 · Zbl 1433.14035
[70] D.O. Orlov, Exceptional set of vector bundles on the variety \[V_5\] V5, Moscow Univ. Math. Bull., 46 (1991), 48-50.
[71] Yu.G. Prokhorov, Automorphism groups of Fano 3-folds, Russian Math. Surveys, 45 (1990), 222-223. · Zbl 0707.14037
[72] Yu.G. Prokhorov, Exotic Fano varieties, Moscow Univ. Math. Bull., 45 (1990), 36-38. · Zbl 0732.14020
[73] Yu.G. Prokhorov, Fano threefolds of genus 12 and compactifications of C3, St. Petersburg Math. J., 3 (1992), 855-864. · Zbl 0790.14038
[74] Yu.G. Prokhorov, p-elementary subgroups of the Cremona group of rank 3, In: Classification of Algebraic Varieties, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2011, pp. 327-338. · Zbl 1221.14015
[75] Yu.G. Prokhorov, Simple finite subgroups of the Cremona group of rank 3, J. Algebraic Geom., 21 (2012), 563-600. · Zbl 1257.14011
[76] Yu.G. Prokhorov, 2-elementary subgroups of the space Cremona group, In: Automorphisms in Birational and Affine Geometry, Springer Proc. Math. Stat., 79, Springer-Verlag, 2014, pp. 215-229. · Zbl 1327.14070
[77] Yu.G. Prokhorov, On G-Fano threefolds, Izv. Ross. Akad. Nauk Ser. Mat., 79 (2015), 159-174. · Zbl 0791.14013
[78] Yu.G. Prokhorov, \[ \mathbb{Q}\] Q-Fano threefolds of index 7, Proc. Steklov Inst. Math., 294 (2016), 139-153. · Zbl 1360.14111
[79] Yu.G. Prokhorov, On the number of singular points of terminal factorial Fano threefolds, Math. Notes, 101 (2017), 1068-1073. · Zbl 1391.14082
[80] Yu.G. Prokhorov and C.A. Shramov, Jordan property for groups of birational selfmaps, Compos. Math., 150 (2014), 2054-2072. · Zbl 1314.14022
[81] Yu.G. Prokhorov and C.A. Shramov, Jordan property for Cremona groups, Amer. J. Math., 138 (2016), 403-418. · Zbl 1343.14010
[82] Yu.G. Prokhorov and C.A. Shramov, p-subgroups in the space Cremona group, Math. Nachr., 10.1002/mana.201700030 (2017). · Zbl 1423.14099
[83] Yu.G. Prokhorov and C.A. Shramov, Finite groups of birational selfmaps of threefolds, preprint, arXiv:1611.00789, to appear in Math. Res. Lett. · Zbl 1314.14022
[84] Yu.G. Prokhorov and C.A. Shramov, Jordan constant for Cremona group of rank 3, Mosc. Math. J., 17 (2017), 457-509. · Zbl 1411.14018
[85] V.V. Przyjalkowski and C.A. Shramov, Double quadrics with large automorphism groups, Proc. Steklov Inst., 294 (2016), 154-175. · Zbl 1375.14146
[86] P.J. Puts, On some Fano-threefolds that are sections of Grassmannians, Nederl. Akad. Wetensch. Indag. Math., 44 (1982), 77-90. · Zbl 0494.14015
[87] Z. Ran, On a theorem of Martens, Rend. Sem. Mat. Univ. Politec. Torino, 44 (1986), 287-291. · Zbl 0632.14027
[88] M. Reid, The complete intersection of two or more quadrics, Ph.D. thesis, Univ. of Cambridge, 1972.
[89] M. Reid, Lines on Fano 3-folds according to Shokurov, Technical Report, 11, Mittag-Leffler Inst., 1980. · Zbl 1352.14029
[90] M. Reid, Minimal models of canonical 3-folds, In: Algebraic Varieties and Analytic Varieties, Tokyo, 1981, Adv. Stud. Pure Math., 1, North-Holland, Amsterdam, 1983, pp. 131-180. · Zbl 1428.14024
[91] F. Russo, On the Geometry of Some Special Projective Varieties, Lect. Notes Unione Mat. Ital., 18, Springer-Verlag, 2016. · Zbl 1337.14001
[92] G. Sanna, Rational curves and instantons on the Fano threefold Y5, preprint, arXiv:1411.7994. · Zbl 0422.14019
[93] Schreyer F.-O.: Geometry and algebra of prime Fano 3-folds of genus 12. Compositio Math. 127, 297-319 (2001) · Zbl 1049.14036
[94] C. Segre, Le superficie degli iperspazi con una doppia infinità di curve piane o spaziali, Atti R. Acc. Sci. Torino, 56 (1920-21), 78-89; Opere. Vol. II, 163-175. · Zbl 0494.14015
[95] I.R. Shafarevich, Basic Algebraic Geometry. 1. Varieties in Projective Space. Translated from the 1988 Russian edition and with notes by Miles Reid. Second ed., Springer-Verlag, 1994. · Zbl 0797.14001
[96] T. Shioda, Arithmetic and geometry of Fermat curves, In: Algebraic Geometry Seminar, Singapore, 1987, World Sci. Publishing, Singapore, 1988, pp. 95-102. · Zbl 1047.14027
[97] Shokurov V.V: The existence of a line on Fano varieties, Izv. Akad. Nauk SSSR Ser. Mat., 43, 922-964 (1979) · Zbl 0422.14019
[98] Tennison B.R: On the quartic threefold. Proc. London Math. Soc. (3) 29, 714-734 (1974) · Zbl 0308.14005
[99] Tolman S.: On a symplectic generalization of Petrie’s conjecture. Trans. Amer. Math. Soc. 362, 3963-3996 (2010) · Zbl 1216.53074
[100] C. Voisin, Hodge Theory and Complex Algebraic Geometry. II. Translated from the French by Leila Schneps, Cambridge Stud. Adv. Math., 77, Cambridge Univ. Press, Cambridge, 2007. · Zbl 1129.14019
[101] Wavrik J.J.: Deformations of Banach coverings of complex manifolds. Amer. J. Math., 90, 926-960 (1968) · Zbl 0176.03902
[102] Yasinsky E.: The Jordan constant for Cremona group of rank 2. Bull. Korean Math. Soc., 54, 1859-1871 (2017) · Zbl 1428.14024
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