Geodesic
Rationality of Fano Threefolds with Terminal Gorenstein Singularities. I
Informatics and Automation, Algebra, number theory, and algebraic geometry, Tome 307 (2019), pp. 230-253.

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We classify some special classes of nonrational Fano threefolds with terminal singularities. In particular, all such hyperelliptic and trigonal varieties are found. 
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Yuri G. Prokhorov. Rationality of Fano Threefolds with Terminal Gorenstein Singularities. I. Informatics and Automation, Algebra, number theory, and algebraic geometry, Tome 307 (2019), pp. 230-253. http://geodesic.mathdoc.fr/item/TRSPY_2019_307_a12/

[1] Alzati A., Bertolini M., “On the rationality of Fano 3-folds with $B_2\ge 2$”, Matematiche, 47:1 (1992), 63–74 | MR | Zbl

[2] Beauville A., “Variétés de Prym et jacobiennes intermédiaires”, Ann. sci. Éc. norm. supér. Sér. 4, 10:3 (1977), 309–391 | MR | Zbl

[3] Blanc J., Lamy S., “On birational maps from cubic threefolds”, North-West. Eur. J. Math., 1 (2015), 69–110 | MR | Zbl

[4] Brown G., Corti A., Zucconi F., “Birational geometry of 3-fold Mori fibre spaces”, The Fano conference, Univ. Torino, Turin, 2004, 235–275 | MR | Zbl

[5] Campana F., Flenner H., “Projective threefolds containing a smooth rational surface with ample normal bundle”, J. reine angew. Math., 440 (1993), 77–98 | MR | Zbl

[6] I. A. Chel'tsov, “Bounded three-dimensional Fano varieties of integer index”, Math. Notes, 66:3 (1999), 360–365 | DOI | DOI | MR | Zbl

[7] Cheltsov I., Park J., “Sextic double solids”, Cohomological and geometric approaches to rationality problems: New perspectives, Progr. Math., 282, Birkhäuser, Boston, MA, 2010, 75–132 | MR | Zbl

[8] Cheltsov I., Przyjalkowski V., Shramov C., “Which quartic double solids are rational?”, J. Algebr. Geom., 28:2 (2019), 201–243 | DOI | MR | Zbl

[9] Clemens C.H., Griffiths P.A., “The intermediate Jacobian of the cubic threefold”, Ann. Math. Ser. 2, 95 (1972), 281–356 | DOI | MR | Zbl

[10] Corti A., Mella M., “Birational geometry of terminal quartic 3-folds. I”, Amer. J. Math., 126:4 (2004), 739–761 | DOI | MR | Zbl

[11] Cutkosky S., “Elementary contractions of Gorenstein threefolds”, Math. Ann., 280:3 (1988), 521–525 | DOI | MR | Zbl

[12] Cutrone J.W., Marshburn N.A., “Towards the classification of weak Fano threefolds with $\rho =2$”, Cent. Eur. J. Math., 11:9 (2013), 1552–1576 | MR | Zbl

[13] M. M. Grinenko, “Birational properties of pencils of del Pezzo surfaces of degrees 1 and 2”, Sb. Math., 191:5 (2000), 633–653 | DOI | DOI | MR | Zbl

[14] M. M. Grinenko, “On a double cone over a Veronese surface”, Izv. Math., 67:3 (2003), 421–438 | DOI | DOI | MR | Zbl

[15] M. M. Grinenko, “Mori structures on a Fano threefold of index 2 and degree 1”, Proc. Steklov Inst. Math., 246 (2004), 103–128 | MR | Zbl

[16] Hassett B., Tschinkel Yu., “On stable rationality of Fano threefolds and del Pezzo fibrations”, J. reine angew. Math., 751 (2019), 275–287 | DOI | MR | Zbl

[17] V. A. Iskovskih, “Fano 3-folds. I”, Math. USSR, Izv., 11:3 (1977), 485–527 | DOI | MR | Zbl

[18] V. A. Iskovskikh, “Anticanonical models of three-dimensional algebraic varieties”, J. Sov. Math., 13 (1980), 745–814 | DOI | MR | Zbl

[19] V. A. Iskovskikh, “Birational automorphisms of three-dimensional algebraic varieties”, J. Sov. Math., 13 (1980), 815–868 | DOI | MR | Zbl

[20] Iskovskikh V.A., “On the rationality problem for conic bundles”, Duke Math. J., 54 (1987), 271–294 | DOI | MR | Zbl

[21] V. A. Iskovskikh and Ju. I. Manin, “Three-dimensional quartics and counterexamples to the Lüroth problem”, Math. USSR, Sb., 15:1 (1971), 141–166 | DOI | MR | Zbl

[22] Iskovskikh V.A., Prokhorov Yu.G., Fano varieties, Encycl. Math. Sci., 47, Algebraic Geometry V, Springer, Berlin, 1999 | MR | Zbl

[23] V. A. Iskovskikh and A. V. Pukhlikov, “Birational automorphisms of multidimensional algebraic manifolds”, Algebraic Geometry–1, Itogi Nauki Tekh., Ser.: Sovrem. Mat. Prilozh., Temat. Obz., 19, VINITI, Moscow, 2001, 5–139 | DOI | MR | Zbl

[24] J. Math. Sci., 82:4 (1996), 3528–3613 | DOI | MR | Zbl

[25] Jahnke P., Peternell T., Radloff I., “Threefolds with big and nef anticanonical bundles. II”, Cent. Eur. J. Math., 9:3 (2011), 449–488 | DOI | MR | Zbl

[26] Jahnke P., Radloff I., “Gorenstein Fano threefolds with base points in the anticanonical system”, Compos. Math., 142:2 (2006), 422–432 | DOI | MR | Zbl

[27] Kaloghiros A.-S., “The defect of Fano 3-folds”, J. Algebr. Geom., 20:1 (2011), 127–149 ; “Errata”, J. Algebr. Geom., 21:2 (2012), 397–399 | DOI | MR | Zbl | DOI | MR | Zbl

[28] Kaloghiros A.-S., “A classification of terminal quartic 3-folds and applications to rationality questions”, Math. Ann., 354:1 (2012), 263–296 | DOI | MR | Zbl

[29] Kawamata Y., “Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces”, Ann. Math. Ser. 2, 127:1 (1988), 93–163 | DOI | MR | Zbl

[30] Kawamata Y., “Small contractions of four dimensional algebraic manifolds”, Math. Ann., 284:4 (1989), 595–600 | DOI | MR | Zbl

[31] Kollár J., Mori S., Birational geometry of algebraic varieties, With the collaboration of C.H. Clemens and A. Corti, Cambridge Tracts Math., 134, Cambridge Univ. Press, Cambridge, 1998 | MR | Zbl

[32] Yu. I. Manin, Cubic Forms: Algebra, Geometry, Arithmetic, North-Holland Math. Libr., 4, North-Holland, Amsterdam, 1974 | MR | MR | Zbl

[33] Mella M., “Existence of good divisors on Mukai varieties”, J. Algebr. Geom., 8:2 (1999), 197–206 | MR | Zbl

[34] Mori S., “Threefolds whose canonical bundles are not numerically effective”, Ann. Math. Ser. 2, 116 (1982), 133–176 | DOI | MR | Zbl

[35] Mori S., Mukai S., “On Fano 3-folds with $B_2\geq 2$”, Algebraic varieties and analytic varieties, Proc. Symp. (Tokyo, 1981), Adv. Stud. Pure Math., 1, North-Holland, Amsterdam, 1983, 101–129 | DOI | MR

[36] Prokhorov Yu.G., “A remark on Fano threefolds with canonical Gorenstein singularities”, The Fano conference, Univ. Torino, Turin, 2004, 647–657 | MR | Zbl

[37] Yu. G. Prokhorov, “On the degree of Fano threefolds with canonical Gorenstein singularities”, Sb. Math., 196:1 (2005), 77–114 | DOI | DOI | MR | Zbl

[38] Prokhorov Yu., “G-Fano threefolds. I”, Adv. Geom., 13:3 (2013), 389–418 | MR | Zbl

[39] Yu. G. Prokhorov, “On $G$-Fano threefolds”, Izv. Math., 79:4 (2015), 795–808 | DOI | DOI | MR | Zbl

[40] Yu. G. Prokhorov, “Singular Fano threefolds of genus 12”, Sb. Math., 207:7 (2016), 983–1009 | DOI | DOI | MR | Zbl

[41] Yu. G. Prokhorov, “On the number of singular points of terminal factorial Fano threefolds”, Math. Notes, 101:6 (2017), 1068–1073 | DOI | DOI | MR | Zbl

[42] Yu. G. Prokhorov, “The rationality problem for conic bundles”, Russ. Math. Surv., 73:3 (2018), 375–456 | DOI | DOI | MR | Zbl

[43] Prokhorov Yu., Reid M., “On $\mathbf Q$-Fano 3-folds of Fano index 2”, Minimal models and extremal rays, Proc. Conf. (RIMS, Kyoto, 2011), Adv. Stud. Pure Math., 70, Math. Soc. Japan, Tokyo, 2016, 397–420 | DOI | MR | Zbl

[44] Prokhorov Yu.G., Shokurov V.V., “Towards the second main theorem on complements”, J. Algebr. Geom., 18:1 (2009), 151–199 | DOI | MR | Zbl

[45] V. V. Przhyjalkowski, I. A. Cheltsov, and K. A. Shramov, “Hyperelliptic and trigonal Fano threefolds”, Izv. Math., 69:2 (2005), 365–421 | DOI | DOI | MR

[46] A. V. Pukhlikov, “Birational automorphisms of a three-dimensional quartic with a quadratic singularity”, Math. USSR, Sb., 63:2 (1989), 457–482 | DOI | MR | Zbl

[47] Reid M., Projective morphisms according to Kawamata, Preprint, Univ. Warwick, Coventry, 1983 | Zbl

[48] Reid M., “Minimal models of canonical 3-folds”, Algebraic varieties and analytic varieties, Proc. Symp. (Tokyo, 1981), Adv. Stud. Pure Math., 1, North-Holland, Amsterdam, 1983, 131–180 | DOI | MR

[49] Saint-Donat B., “Projective models of $K$-3 surfaces”, Amer. J. Math., 96 (1974), 602–639 | DOI | MR | Zbl

[50] V. G. Sarkisov, “On conic bundle structures”, Math. USSR, Izv., 20:2 (1983), 355–390 | DOI | MR | Zbl

[51] Shin K.-H., “3-dimensional Fano varieties with canonical singularities”, Tokyo J. Math., 12:2 (1989), 375–385 | DOI | MR | Zbl

[52] V. V. Šokurov, “Smoothness of the general anticanonical divisor on a Fano 3-fold”, Math. USSR, Izv., 14:2 (1980), 395–405 | DOI | MR | MR | Zbl

[53] V. V. Shokurov, “Prym varieties: Theory and applications”, Math. USSR, Izv., 23:1 (1984), 83–147 | DOI | MR

[54] Takeuchi K., Weak Fano threefolds with del Pezzo fibration, E-print, 2009, arXiv: 0910.2188 [math.AG]

[55] A. N. Tyurin, “The middle Jacobian of three-dimensional varieties”, J. Sov. Math., 13 (1980), 707–745 | DOI | MR | Zbl

[56] Voisin C., “Sur la jacobienne intermédiaire du double solide d'indice deux”, Duke Math. J., 57:2 (1988), 629–646 | DOI | MR | Zbl