×

zbMATH — the first resource for mathematics

Super-rigid affine Fano varieties. (English) Zbl 1408.14052
Motivated by the study of automorphsm groups of affine varieties and the study of birational super-rigidity of Fano varieties, this paper studies affine Fano varieties and their super-rigidity.
This paper defines affine Fano varieties as the following: \(X\setminus S_X\) is an affine Fano variety with completion \(X\) and boundary \(S_X\) if \(X\) is a \(\mathbb{Q}\)-factorial projective normal variety of Picard rank \(1\) and \(S_X\) is a prime divisor on \(X\) such that \((X, S_X)\) is purely log terminal and \(-(K_X+S_X)\) is ample. Then super-rigidity for affine Fano varieties is defined as an analogue of birational super-rigidity of Fano varieties.
As the first result, this paper provides a sufficient condition for affine Fano varieties to be super-rigid, that is, \(X\setminus S_X\) is super-rigid if for any mobile linear system \(\mathcal{M}_X\) on \(X\) and a positive rational number \(\lambda\) such that \(K_X+S_X+\lambda \mathcal{M}_X\sim_{\mathbb{Q}} 0\) the pair \((X, S_X+\lambda \mathcal{M}_X)\) is log canonical along \(S_X\) This is an analogue of the Noether-Fano inequality.
Another sufficient condition is provided by using the alpha-invariants, which says that \(X\setminus S_X\) is super-rigid if \(\alpha(S_X, \text{Diff}_{S_X}(0))\geq 1\).
With those conditions, many examples and non-examples are provided.

MSC:
14E07 Birational automorphisms, Cremona group and generalizations
14J45 Fano varieties
14J50 Automorphisms of surfaces and higher-dimensional varieties
14R10 Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem)
14R20 Group actions on affine varieties
14R25 Affine fibrations
14C20 Divisors, linear systems, invertible sheaves
14E05 Rational and birational maps
14J17 Singularities of surfaces or higher-dimensional varieties
14J70 Hypersurfaces and algebraic geometry
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Blanc, J.; Dubouloz, A., Automorphisms of A1 -fibered surfaces, Trans. Amer. Math. Soc., 363, 5887-5924, (2011) · Zbl 1239.14053
[2] Blanc, J.; Dubouloz, A., Affine surfaces with a huge group of automorphisms, Int. Math. Res. Not. IMRN, 2015, 422-459, (2015) · Zbl 1320.14072
[3] Cassels, J.; Guy, M., On the Hasse principle for cubic surfaces, Mathematika, 13, 111-120, (1966) · Zbl 0151.03405
[4] Cheltsov, I., Birationally rigid Fano varieties, Russian Math. Surveys, 60, 875-965, (2005) · Zbl 1145.14032
[5] Cheltsov, I., Log canonical thresholds of del Pezzo surfaces, Geom. Funct. Anal., 11, 1118-1144, (2008) · Zbl 1161.14030
[6] Cheltsov, I., Fano varieties with many selfmaps, Adv. Math., 217, 97-124, (2008) · Zbl 1126.14050
[7] Cheltsov, I., Log canonical thresholds of Fano threefold hypersurfaces, Izv. Math., 73, 727-795, (2009) · Zbl 1181.14047
[8] Cheltsov, I.; Kosta, D., Computing 𝛼-invariants of singular del Pezzo surfaces, J. Geom. Anal., 24, 798-842, (2014) · Zbl 1309.14031
[9] Cheltsov, I.; Park, J., Birationally rigid Fano threefold hypersurfaces, Mem. Amer. Math. Soc., 246, (2017)
[10] Cheltsov, I.; Park, J.; Shramov, C., Exceptional del Pezzo hypersurfaces, J. Geom. Anal., 20, 787-816, (2010) · Zbl 1211.14047
[11] Cheltsov, I.; Park, J.; Won, J., Affine cones over smooth cubic surfaces, J. Eur. Math. Soc., 18, 1537-1564, (2016) · Zbl 1386.14068
[12] Cheltsov, I.; Park, J.; Won, J., Cylinders in singular del Pezzo surfaces, Compos. Math., 152, 1198-1224, (2016) · Zbl 1360.14020
[13] Cheltsov, I.; Przyjalkowski, V.; Shramov, C.
[14] Cheltsov, I.; Shramov, C., On exceptional quotient singularities, Geom. Topol., 15, 1843-1882, (2011) · Zbl 1232.14001
[15] Cheltsov, I.; Shramov, C., Del Pezzo zoo, Exp. Math., 22, 313-326, (2013) · Zbl 1281.14034
[16] Cheltsov, I.; Shramov, C., Weakly-exceptional singularities in higher dimensions, J. Reine Angew. Math., 689, 201-241, (2014)
[17] Clemens, C.; Griffiths, P., The intermediate Jacobian of the cubic threefold, Ann. of Math. (2), 95, 281-356, (1972) · Zbl 0214.48302
[18] Corti, A., Singularities of linear systems and 3-fold birational geometry, Explicit birational geometry of 3-folds, 259-312, (2000), Cambridge University Press
[19] Corti, A.; Pukhlikov, A.; Reid, M., Fano 3-fold hypersurfaces, Explicit birational geometry of 3-folds, 175-258, (2000), Cambridge University Press
[20] Dubouloz, A.; Kishimoto, T., Log-uniruled affine varieties without cylinder-like open subsets, Bull. Soc. Math. France, 143, 383-401, (2015) · Zbl 1327.14196
[21] Dubouloz, A.; Kishimoto, T., Cylinder in del Pezzo fibrations, Israel J. Math., 225, 797-815, (2018) · Zbl 06898294
[22] Dubouloz, A.; Lamy, S., Automorphisms of open surfaces with irreducible boundary, Osaka J. Math., 52, 747-791, (2015)
[23] De Fernex, T., Birationally rigid hypersurfaces, Invent. Math., 192, 533-566, (2013) · Zbl 1279.14019
[24] Iano-Fletcher, A., Working with weighted complete intersections, Explicit birational geometry of 3-folds, 101-173, (2000), Cambridge University Press
[25] Gizatullin, M., Affine surfaces that can be augmented by a nonsingular rational curve, Izv. Akad. Nauk SSSR Ser. Mat., 34, 778-802, (1970)
[26] Gizatullin, M.; Danilov, V., Automorphisms of affine surfaces. I, Izv. Akad. Nauk SSSR Ser. Mat., 39, 523-565, (1975)
[27] Gizatullin, M.; Danilov, V., Automorphisms of affine surfaces. II, Izv. Akad. Nauk SSSR Ser. Mat., 41, 54-103, (1977)
[28] Grinenko, M., On a double cone over a Veronese surface, Izv. Math., 67, 421-438, (2003) · Zbl 1082.14015
[29] Grinenko, M., Mori structures on a Fano threefold of index 2 and degree 1, Proc. Steklov Inst. Math., 246, 103-128, (2004)
[30] Iskovskih, V.; Manin, Yu., Three-dimensional quartics and counterexamples to the LĂĽroth problem, Mat. Sb., 86, 140-166, (1971)
[31] Johnson, J.; Kollár, J., Fano hypersurfaces in weighted projective 4-spaces, Exp. Math., 10, 151-158, (2001) · Zbl 0972.14034
[32] Keel, S.; Mckernan, J., Rational curves on quasi-projective surfaces, Mem. Amer. Math. Soc., 140, (1999)
[33] Kishimoto, T.; Prokhorov, Yu.; Zaidenberg, M.
[34] Kishimoto, T.; Prokhorov, Yu.; Zaidenberg, M., G_a -actions on affine cones, Transform. Groups, 18, 1137-1153, (2013) · Zbl 1297.14061
[35] Kollár, J., Singularities of pairs, Algebraic geometry (Santa Cruz, 1995) Part 1, 221-287, (1997), American Mathematical Society
[36] Miyanishi, M.; Sugie, T., Homology planes with quotient singularities, J. Math. Kyoto Univ., 31, 755-788, (1991) · Zbl 0790.14034
[37] Prokhorov, Y.; Zaidenberg, M., Examples of cylindrical Fano fourfolds, Eur. J. Math., 2, 262-282, (2016) · Zbl 1375.14206
[38] Pukhlikov, A., Birational automorphisms of Fano hypersurfaces, Invent. Math., 134, 401-426, (1998) · Zbl 0964.14011
[39] Pukhlikov, A., Birational geometry of Fano direct products, Izv. Math., 69, 1225-1255, (2005) · Zbl 1119.14011
[40] Pukhlikov, A., Birationally rigid varieties, (2013), American Mathematical Society: American Mathematical Society, Providence, RI
[41] Pukhlikov, A., Automorphisms of certain affine complements in the projective space, Sb. Math., 209, 276-289, (2018) · Zbl 1439.14052
[42] Rosenlicht, M., Some basic theorems on algebraic groups, Amer. J. Math., 78, 401-443, (1956) · Zbl 0073.37601
[43] Russell, P., On affine-ruled rational surfaces, Math. Ann., 255, 287-302, (1981) · Zbl 0438.14024
[44] Sakovics, D., Weakly-exceptional quotient singularities, Cent. Eur. J. Math., 10, 885-902, (2012) · Zbl 1263.14007
[45] Sakovics, D., Five-dimensional weakly exceptional quotient singularities, Proc. Edinb. Math. Soc. (2), 57, 269-279, (2014) · Zbl 1306.14003
[46] Shokurov, V., Three-fold log flips, Izv. Math., 40, 95-202, (1993) · Zbl 0785.14023
[47] Szurek, M.; Wiśniewski, J., Fano bundles of rank 2 on surfaces, Compos. Math., 76, 295-305, (1990)
[48] Tian, G., On Käahler-Einstein metrics on certain Kähler manifolds with C_1(M) > 0, Invent. Math., 89, 225-246, (1987) · Zbl 0599.53046
[49] Voisin, C., Sur la jacobienne intermĂ©diaire du double solide d’indice deux, Duke Math. J., 57, 629-646, (1988) · Zbl 0698.14049
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.