Geodesic
Birationally Rigid Finite Covers of the Projective Space
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algebra, number theory, and algebraic geometry, Tome 307 (2019), pp. 254-266.

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In this paper we prove birational superrigidity of finite covers of degree $d$ of the $M$-dimensional projective space of index $1$, where $d\geq 5$ and $M\geq 10$, that have at most quadratic singularities of rank ${\geq }\,7$ and satisfy certain regularity conditions. Up to now, only cyclic covers have been studied in this respect. The set of varieties that have worse singularities or do not satisfy the regularity conditions is of codimension ${\geq }\,(M-4)(M-5)/2+1$ in the natural parameter space of the family.
Keywords: maximal singularity, linear system, Fano variety, self-intersection, hypertangent divisor.
Mots-clés : birational map 
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A. V. Pukhlikov. Birationally Rigid Finite Covers of the Projective Space. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algebra, number theory, and algebraic geometry, Tome 307 (2019), pp. 254-266. http://geodesic.mathdoc.fr/item/TM_2019_307_a13/

[1] Call F., Lyubeznik G., “A simple proof of Grothendieck's theorem on the parafactoriality of local rings”, Commutative algebra: Syzygies, multiplicities, and birational algebra, AMS–IMS–SIAM Summer Res. Conf. (1992), Contemp. Math., 159, Amer. Math. Soc., Providence, RI, 1994, 15–18 | DOI | MR | Zbl

[2] I. A. Cheltsov, “Double space with double line”, Sb. Math., 195:10 (2004), 1503–1544 | DOI | DOI | MR | Zbl

[3] I. A. Cheltsov, “Birationally superrigid cyclic triple spaces”, Izv. Math., 68:6 (2004), 1229–1275 | DOI | DOI | MR | Zbl

[4] Cheltsov I., “On nodal sextic fivefold”, Math. Nachr., 280:12 (2007), 1344–1353 | DOI | MR | Zbl

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

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

[7] V. A. Iskovskih and Ju. I. Manin, “Three-dimensional quartics and counterexamples to the Lüroth problem”, Math. USSR, Sb., 86:1 (1971), 140–166 | MR | Zbl

[8] R. Mullany, “Fano double spaces with a big singular locus”, Math. Notes, 87:3 (2010), 444–448 | DOI | DOI | MR | Zbl

[9] A. V. Pukhlikov, “Birational automorphisms of a double space and double quadric”, Math. USSR, Izv., 32:1 (1989), 233–243 | DOI | MR | Zbl

[10] A. V. Pukhlikov, “Birational automorphisms of double spaces with singularities”, Algebraic Geometry–2, Itogi Nauki Tekh., Ser.: Sovrem. Mat. Prilozh., Temat. Obz., 24, VINITI, Moscow, 2001, 177–196 | DOI | MR | Zbl

[11] J. Math. Sci., 85:4 (1997), 2128–2141 | DOI | MR | Zbl

[12] Pukhlikov A.V., “Birational geometry of algebraic varieties with a pencil of Fano cyclic covers”, Pure Appl. Math. Q., 5:2 (2009), 641–700 | DOI | MR | Zbl

[13] Pukhlikov A., Birationally rigid varieties, Math. Surv. Monogr., 190, Am. Math. Soc., Providence, RI, 2013 | DOI | MR | Zbl

[14] A. V. Pukhlikov, “Birationally rigid Fano fibre spaces. II”, Izv. Math., 79:4 (2015), 809–837 | DOI | DOI | MR | Zbl

[15] Pukhlikov A.V., “The $4n^2$-inequality for complete intersection singularities”, Arnold Math. J., 3:2 (2017), 187–196 | DOI | MR | Zbl