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Fano threefolds as equivariant compactifications of the vector group. (English) Zbl 1455.14071
Summary: In this article, we determine all equivariant compactifications of the three-dimensional vector group $$\mathbf{G}_a^3$$ that are smooth Fano threefolds with Picard number greater than or equal to two.
##### MSC:
 14J10 Families, moduli, classification: algebraic theory 14J45 Fano varieties 14G05 Rational points 14M17 Homogeneous spaces and generalizations 14M27 Compactifications; symmetric and spherical varieties
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##### References:
 [1] [AB92] A. Alzati and M. Bertolini, On the rationality of Fano 3-folds with $$B_2{\ge }2$$, Matematiche 47 (1992), no. 1, 63-74. · Zbl 0787.14023 [2] [AP14] I. Arzhantsev and A. Popovskiy, Additive actions on projective hypersurfaces, Automorphisms in birational and affine geometry. Papers based on the presentations at the conference, Levico Terme, Italy, October 29-November 3, 2012, pp. 17-33, Springer, Cham, 2014. · Zbl 1326.14112 [3] [AR17] I. Arzhantsev and E. Romaskevich, Additive actions on toric varieties, Proc. Amer. Math. Soc. 145 (2017), no. 5, 1865-1879. · Zbl 1375.14155 [4] [Arz11] I. V. Arzhantsev, Flag varieties as equivariant compactifications of $$\mathbb{G}_a^n$$, Proc. Amer. Math. Soc. 139 (2011), no. 3, 783-786. · Zbl 1217.14032 [5] [AS11] I. V. Arzhantsev and E. V. Sharoyko, Hassett-tschinkel correspondence: modality and projective hypersurfaces, J. Algebra 348 (2011), no. 1, 217-232. · Zbl 1248.14053 [6] [Bat82] V. V. Batyrev, Toroidal Fano 3-folds, Math. USSR, Izv. 19 (1982), 13-25. · Zbl 0495.14027 [7] [Bil17] M. Bilu, Produits eulériens motiviques, Ph.D. thesis, Université Paris Saclay, 2017. [8] [Bir16a] C. Birkar, Anti-pluricanonical systems on Fano varieties. Ann. of Math. (2019, to appear), arXiv:1603.05765. · Zbl 07107180 [9] [Bir16b] C. Birkar, Singularities of linear systems and boundedness of Fano varieties, arXiv preprint, 2016, arXiv:1609.05543. [10] [Bla56] A. Blanchard, Sur les variétés analytiques complexes, Ann. Sci. Éc. Norm. Supér. 3 (1956), no. 73, 157-202. · Zbl 0073.37503 [11] [BM78] L. Brenton and J. Morrow, Compactifications of $$\mathbf{C}^n$$, Trans. Amer. Math. Soc. 246 (1978), 139-153. · Zbl 0416.32015 [12] [Bri17] M. Brion, Some structure theorems for algebraic groups, Algebraic groups: structure and actions, Proc. Sympos. Pure Math., 94, pp. 53-126, Amer. Math. Soc., Providence, RI, 2017. · Zbl 1401.14195 [13] [CLT02] A. Chambert-Loir and Y. Tschinkel, On the distribution of points of bounded height on equivariant compactifications of vector groups, Invent. Math. 148 (2002), no. 2, 421-452. · Zbl 1067.11036 [14] [CLT12] A. Chambert-Loir and Y. Tschinkel, Integral points of bounded height on partial equivariant compactifications of vector groups, Duke Math. J. 161 (2012), no. 15, 2799-2836. · Zbl 1348.11055 [15] [CLS11] D. A. Cox, J. B. Little, and H. K. Schenck, Toric varieties, American Mathematical Society (AMS), Providence, RI, 2011. · Zbl 1223.14001 [16] [Dem70] M. Demazure, Sous-groupes algébriques de rang maximum du groupe de Cremona, Ann. Sci. Éc. Norm. Supér. 4 (1970), no. 3, 507-588. · Zbl 0223.14009 [17] [DL10] U. Derenthal and D. Loughran, Singular del Pezzo surfaces that are equivariant compactifications, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 377 (2010), no. 10, 26-43. · Zbl 1296.14016 [18] [DL15] U. Derenthal and D. Loughran, Equivariant compactifications of two-dimensional algebraic groups, Proc. Edinb. Math. Soc. (2) 58 (2015), no. 1, 149-168. · Zbl 1368.14059 [19] [Dev15] R. Devyatov, Unipotent commutative group actions on flag varieties and nilpotent multiplications, Transform. Groups 20 (2015), no. 1, 21-64. · Zbl 1393.14047 [20] [Dol12] I. V. Dolgachev, Classical algebraic geometry. A modern view, Cambridge University Press, Cambridge, 2012. · Zbl 1252.14001 [21] [FH14] B. Fu and J.-M. Hwang, Uniqueness of equivariant compactifications of $$\mathbb{C}^n$$ by a Fano manifold of Picard number 1, Math. Res. Lett. 21 (2014), no. 1, 121-125. · Zbl 1327.32030 [22] [FH17] B. Fu and J.-M. Hwang, Euler-symmetric projective varieties, 2017, arXiv:1707.06764. [23] [Fur86] M. Furushima, Singular del Pezzo surfaces and analytic compactifications of $$3$$-dimensional complex affine space $$\mathbf{C}^3$$, Nagoya Math. J. 104 (1986), 1-28. · Zbl 0612.14037 [24] [Fur90] M. Furushima, Complex analytic compactifications of $$\mathbf{C}^3$$, Compos. Math. 76 (1990), no. 1-2, 163-196. · Zbl 0721.32012 [25] [Fur93a] M. Furushima, The complete classification of compactifications of $$\mathbf{C}^3$$ which are projective manifolds with the second Betti number one, Math. Ann. 297 (1993a), no. 4, 627-662. · Zbl 0788.32022 [26] [Fur93b] M. Furushima, A new example of a compactification of $$\mathbf{C}^3$$, Math. Z. 212 (1993b), no. 3, 395-399. · Zbl 0790.32027 [27] [FN89a] M. Furushima and N. Nakayama, The family of lines on the Fano threefold $$V_5$$, Nagoya Math. J. 116 (1989a), 111-122. · Zbl 0731.14025 [28] [FN89b] M. Furushima and N. Nakayama, A new construction of a compactification of $$\mathbf{C}^3$$, Tohoku Math. J. (2) 41 (1989b), no. 4, 543-560. · Zbl 0703.14025 [29] [Har70] R. Hartshorne, Ample subvarieties of algebraic varieties, Lecture Notes in Math., 156, Springer-Verlag, Berlin-New York, 1970, Notes written in collaboration with C. Musili. · Zbl 0208.48901 [30] [Har77] R. Hartshorne, Algebraic geometry, Grad. Texts in Math., 52, Springer-Verlag, New York-Heidelberg, 1977. [31] [HT99] B. Hassett and Yu. Tschinkel, Geometry of equivariant compactifications of $$\mathbb{G}^n_a$$, Int. Math. Res. Not. 1999 (1999), no. 22, 1211-1230. · Zbl 0966.14033 [32] [Hir54] F. Hirzebruch, Some problems on differentiable and complex manifolds, Ann. of Math. (2) 60 (1954), 213-236. · Zbl 0056.16803 [33] [Isk77] V. A. Iskovskih, Fano threefolds. I, Izv. Ross. Akad. Nauk Ser. Mat. 41 (1977), no. 3, 516-562, 717. [34] [Isk78] V. A. Iskovskih, Fano threefolds. II, Izv. Ross. Akad. Nauk Ser. Mat. 42 (1978), no. 3, 506-549. [35] [Isk79] V. A. Iskovskih, Anticanonical models of three-dimensional algebraic varieties, Current problems in mathematics, vol. 12 (Russian), 239 (loose errata), pp. 59-157, VINITI, Moscow, 1979. [36] [IP99] V. A. Iskovskikh and Yu. G. Prokhorov, Fano varieties, Algebraic geometry V: Fano varieties. Transl. from the Russian by Yu. G. Prokhorov and S. Tregub, pp. 1-245, Springer, Berlin, 1999. [37] [Keb98] S. Kebekus, Simple models of quasihomogeneous projective $$3$$-folds, Doc. Math. 3 (1998), 15-26. · Zbl 0940.14029 [38] [Kis02] T. Kishimoto, A new proof of a theorem of Ramanujam-Morrow, J. Math. Kyoto Univ. 42 (2002), no. 1, 117-139. · Zbl 1035.14023 [39] [Kis05] T. Kishimoto, Compactifications of contractible affine 3-folds into smooth Fano 3-folds with $$B_2=2$$, Math. Z. 251 (2005), no. 4, 783-820. · Zbl 1086.14050 [40] [Kod71] K. Kodaira, Holomorphic mappings of polydiscs into compact complex manifolds, J. Differential Geom. 6 (1971), 33-46. · Zbl 0227.32008 [41] [KMM92] J. Kollár, Y. Miyaoka, and S. Mori, Rational connectedness and boundedness of Fano manifolds, J. Differential Geom. 36 (1992), no. 3, 765-779. · Zbl 0759.14032 [42] [KM98] J. Kollár and S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Math., 134, Cambridge University Press, Cambridge, 1998, With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original.. [43] [LM09] R. Lazarsfeld and M. Mustaţă, Convex bodies associated to linear series, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), no. 5, 783-835. · Zbl 1182.14004 [44] [Mor82] S. Mori, Threefolds whose canonical bundles are not numerically effective, Ann. of Math. (2) 116 (1982), no. 1, 133-176. · Zbl 0557.14021 [45] [MM81] S. Mori and S. Mukai, Classification of Fano 3-folds with $$B_2\ge 2$$, Manuscripta Math. 36 (1981), 147-162. · Zbl 0478.14033 [46] [MM03] S. Mori and S. Mukai, Erratum: “Classification of Fano 3-folds with $$B_2\ge 2$$” [Manuscripta Math. 36 (1981/82), no. 2, 147-162], Manuscripta Math. 110 (2003), 407. [47] [Mor72] J. A. Morrow, Compactifications of $$\mathbf{C}^2$$, Bull. Amer. Math. Soc. 78 (1972), 813-816. · Zbl 0257.32013 [48] [Mor73] J. A. Morrow, Minimal normal compactifications of $$\mathbf{C^2}$$, Rice Univ. Studies 59 (1973), no. 1, 97-112. Complex analysis, 1972 (Proc. Conf., Rice Univ., Houston, Tex., 1972), Vol. I: Geometry of singularities. [49] [Muk92] S. Mukai, Fano $$3$$-folds, Complex projective geometry, Trieste, 1989/Bergen 1989, London Math. Soc. Lecture Note Ser., 179, pp. 255-263, Cambridge Univ. Press, Cambridge, 1992. [50] [MS90] S. Müller-Stach, Compactification of $${\mathbb{C}}^3$$ with reducible boundary divisor, Math. Ann. 286 (1990), no. 1-3, 409-431. · Zbl 0681.32020 [51] [Nag18] M. Nagaoka, Fano compactifications of contractible affine 3-folds with trivial log canonical divisors, Internat. J. Math. 29 (2018), no. 6, 1850042. · Zbl 1391.14117 [52] [Pet89] T. Peternell, Compactifications of $$\mathbf{C}^3$$. II, Math. Ann. 283 (1989), no. 1, 121-137. [53] [Pet90] T. Peternell, Compactifications of $$\mathbf{C}^3$$. III, Math. Z. 205 (1990), no. 2, 213-222. · Zbl 0732.32015 [54] [PS88] T. Peternell and M. Schneider, Compactifications of $$\mathbf{C}^3$$. I, Math. Ann. 280 (1988), no. 1, 129-146. · Zbl 0651.14025 [55] [Pey02] E. Peyre, Points de hauteur bornée et géométrie des variétés (d’après Y. Manin et al.), Astérisque 282 (2002), no. 891, 323-344. · Zbl 1039.11045 [56] [Poo08] B. Poonen, Isomorphism types of commutative algebras of finite rank over an algebraically closed field, Computational arithmetic geometry, Contemp. Math., 463, pp. 111-120, Amer. Math. Soc., Providence, RI, 2008. · Zbl 1155.13015 [57] [PZ18] Yu. Prokhorov and M. Zaidenberg, Fano-Mukai fourfolds of genus 10 as compactifications of $$\mathbb{C}^4$$, Eur. J. Math. 4 (2018), no. 3, 1197-1263. · Zbl 1423.14244 [58] [Pro91] Yu. G. Prokhorov, Fano threefolds of genus $$12$$ and compactifications of $$\mathbf{C}^3$$, Algebra i Analiz 3 (1991), no. 4, 162-170. [59] [Pro94] Yu. G. Prokhorov, Compactifications of $$\mathbf{C}^4$$ of index $$3$$, Algebraic geometry and its applications, Yaroslavl, 1992, Aspects Math., E25, pp. 159-169, Friedr. Vieweg, Braunschweig, 1994. [60] [RvdV60] R. Remmert and T. van de Ven, Zwei Sätze über die komplex-projektive Ebene, Nieuw Arch. Wiskd. (5) 3 (1960), no. 8, 147-157. · Zbl 0136.20703 [61] [RRS92] R. Richardson, G. Röhrle, and R. Steinberg, Parabolic subgroups with Abelian unipotent radical, Invent. Math. 110 (1992), no. 3, 649-671. · Zbl 0786.20029 [62] [Sha09] E. V. Sharoyko, Hassett-Tschinkel correspondence and automorphisms of a quadric, Sb. Math. 200 (2009), no. 11, 145-160. [63] [Sho79] V. V. Shokurov, The existence of a line on Fano varieties, Izv. Ross. Akad. Nauk Ser. Mat. 43 (1979), no. 4, 922-964, 968. · Zbl 0422.14019 [64] [Sum74] H. Sumihiro, Equivariant completion, J. Math. Kyoto Univ. 14 (1974), 1-28. · Zbl 0277.14008 [65] [Sup56] D. A. Suprunenko, On maximal commutative subalgebras of the full linear algebra, Uspekhi Mat. Nauk 11 (1956), no. 3, 181-184. [66] [WW82] K. · Zbl 0581.14028
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