×

Quartic double solids with icosahedral symmetry. (English) Zbl 1430.14084

Summary: We study quartic double solids admitting icosahedral symmetry.

MSC:

14J45 Fano varieties
14E05 Rational and birational maps
14E07 Birational automorphisms, Cremona group and generalizations
14E08 Rationality questions in algebraic geometry
14J50 Automorphisms of surfaces and higher-dimensional varieties
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Artin, M., Mumford, D.: Some elementary examples of unirational varieties which are not rational. Proc. London Math. Soc. s3-25(1), 75-95 (1972) · Zbl 0244.14017
[2] Beauville, A.; Faber, C. (ed.); Farkas, G. (ed.); de Jong, R. (ed.), Non-rationality of the symmetric sextic Fano threefold, 57-60 (2012), Zürich · Zbl 1317.14033
[3] Blichfeldt, H.F.: Finite Collineation Groups. University of Chicago Press, Chicago (1917)
[4] Cheltsov, I.A.: Birationally rigid Fano varieties. Russian Math. Surveys 60(5), 875-965 (2005) · Zbl 1145.14032
[5] Cheltsov, I., Przyjalkowski, V., Shramov, C.: Which quartic double solids are rational? (2015). arXiv:1508.07277 · Zbl 1430.14084
[6] Cheltsov, I., Shramov, C.: Three embeddings of the Klein simple group into the Cremona group of rank three. Transform. Groups 17(2), 303-350 (2012) · Zbl 1272.14013
[7] Cheltsov, I., Shramov, C.: Five embeddings of one simple group. Trans. Amer. Math. Soc. 366(3), 1289-1331 (2014) · Zbl 1291.14060
[8] Cheltsov, I., Shramov, C.: Cremona Groups and the Icosahedron. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton (2015) · Zbl 1328.14003
[9] Clemens, C.H.: Double solids. Adv. Math. 47(2), 107-230 (1983) · Zbl 0509.14045
[10] Degtyarev, A.I.: Classification of surfaces of degree four having a nonsimple singular point. Math. USSR-Izv. 35(3), 607-627 (1990) · Zbl 0722.14019
[11] Dolgachev, IV; Iskovskikh, VA; Tschinkel, Yu (ed.); Zarhin, Yu (ed.), Finite subgroups of the plane Cremona group, No. 269, 443-548 (2009), Boston · Zbl 1219.14015
[12] Endraß, S.: On the divisor class group of double solids. Manuscripta Math. 99(3), 341-358 (1999) · Zbl 0970.14006
[13] Hashimoto, K.: Period map of a certain \[K3\] K3 family with an \[{\rm S}_5\] S5-action. J. Reine Angew. Math. 652, 1-65 (2011) · Zbl 1213.14070
[14] Prokhorov, Yu.G.: Fields of invariants of finite linear groups. In: Bogomolov, F., Tschinkel, Yu. (eds.) Cohomological and Geometric Approaches to Rationality Problems. Progress in Mathematics, vol. 282, pp. 245-273. Birkhäuser, Boston (2010) · Zbl 1204.14007
[15] Prokhorov, Yu.: Simple finite subgroups of the Cremona group of rank \[33\]. J. Algebraic Geom. 21(3), 563-600 (2012) · Zbl 1257.14011
[16] Prokhorov, Yu.: \[G\] G-Fano threefolds. I. Adv. Geom. 13(3), 389-418 (2013) · Zbl 1291.14024
[17] Reichstein, Z., Youssin, B.: Essential dimensions of algebraic groups and a resolution theorem for \[G\] G-varieties. Canad. J. Math. 52(5), 1018-1056 (2000) · Zbl 1044.14023
[18] Shokurov, V.V.: Prym varieties: theory and applications. Math. USSR-Izv. 23(1), 83-147 (1984) · Zbl 0572.14025
[19] Shokurov, \[V.: 33\]-fold log flips. Math. USSR-Izv. 40(1), 95-202 (1993) · Zbl 0785.14023
[20] Szurek, M., Wiśniewski, J.A.: Fano bundles of rank \[22\] on surfaces. Compositio Math. 76(1-2), 295-305 (1990) · Zbl 0719.14028
[21] Tikhomirov, A.S.: Singularities of the theta-divisor of the intermediate Jacobian of the double \[{\mathbb{P}}^3\] P3 of index two. Math. USSR-Izv. 21(2), 355-373 (1983) · Zbl 0534.14024
[22] Umezu, Y.: On normal projective surfaces with trivial dualizing sheaf. Tokyo J. Math. 4(2), 343-354 (1981) · Zbl 0496.14025
[23] Voisin, C.: Sur la jacobienne intermédiaire du double solide d’indice deux. Duke Math. J. 57(2), 629-646 (1988) · Zbl 0698.14049
[24] Zagorskii, A.A.: Three-dimensional conical fibrations. Math. Notes 21(6), 420-427 (1977) · Zbl 0399.14026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.