Geodesic
Smooth Prime Fano Complete Intersections in Toric Varieties
Matematičeskie zametki, Tome 109 (2021) no. 4, pp. 590-596.

Voir la notice de l'article provenant de la source Math-Net.Ru

We prove that a smooth well-formed Picard rank-one Fano complete intersection of dimension at least $2$ in a toric variety is a weighted complete intersection.
Keywords: Fano manifold, weighted complete intersection, toric variety. 
@article{MZM_2021_109_4_a9,
     author = {V. V. Przyjalkowski and {\CYRS}. A. Shramov},
     title = {Smooth {Prime} {Fano} {Complete} {Intersections} in {Toric} {Varieties}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {590--596},
     publisher = {mathdoc},
     volume = {109},
     number = {4},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2021_109_4_a9/}
}
TY  - JOUR
AU  - V. V. Przyjalkowski
AU  - С. A. Shramov
TI  - Smooth Prime Fano Complete Intersections in Toric Varieties
JO  - Matematičeskie zametki
PY  - 2021
SP  - 590
EP  - 596
VL  - 109
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2021_109_4_a9/
LA  - ru
ID  - MZM_2021_109_4_a9
ER  - 
%0 Journal Article
%A V. V. Przyjalkowski
%A С. A. Shramov
%T Smooth Prime Fano Complete Intersections in Toric Varieties
%J Matematičeskie zametki
%D 2021
%P 590-596
%V 109
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2021_109_4_a9/
%G ru
%F MZM_2021_109_4_a9
V. V. Przyjalkowski; С. A. Shramov. Smooth Prime Fano Complete Intersections in Toric Varieties. Matematičeskie zametki, Tome 109 (2021) no. 4, pp. 590-596. http://geodesic.mathdoc.fr/item/MZM_2021_109_4_a9/

[1] O. Küchle, “On Fano 4-fold of index $1$ and homogeneous vector bundles over Grassmannians”, Math. Z., 218:4 (1995), 563–575 | DOI | MR | Zbl

[2] O. Küchle, “Some remarks and problems concerning the geography of Fano $4$-folds of index and Picard number one”, Quaestiones Math., 20:1 (1997), 45–60 | DOI | MR | Zbl

[3] E. Fatighenti, G. Mongardi, A note on a Griffiths-type ring for complete intersections in Grassmannians, 2018, arXiv: 1801.09586

[4] J.-J. Chen, J. Chen, M. Chen, “On quasismooth weighted complete intersections”, J. Algebraic Geom., 20:2 (2011), 239–262 | DOI | MR | Zbl

[5] M. Pizzato, T. Sano, L. Tasin, “Effective nonvanishing for Fano weighted complete intersections”, Algebra Number Theory, 11:10 (2017), 2369–2395 | DOI | MR | Zbl

[6] V. V. Przhiyalkovskii, K. A. Shramov, “Avtomorfizmy vzveshennykh polnykh peresechenii”, Algebra, teoriya chisel i algebraicheskaya geometriya, Tr. MIAN, 307, MIAN, M., 2019, 217–229 | DOI | MR

[7] V. Przyjalkowski, C. Shramov, “Bounds for smooth Fano weighted complete intersections”, Commun. Number Theory Phys., 14:3 (2020), 511–553 | MR | Zbl

[8] V. V. Przhiyalkovskii, K. A. Shramov, “Vzveshennye polnye peresecheniya Fano bolshoi korazmernosti”, Sib. matem. zhurn., 61:2 (2020), 377–384 | DOI | MR

[9] V. V. Przhiyalkovskii, K. A. Shramov, “Ob avtomorfizmakh kvazigladkikh vzveshennykh polnykh peresechenii”, Matem. sb., 212:3 (2021) (to appear) ; (2020), arXiv: 2006.01213 | DOI

[10] D. Cox, “The homogeneous coordinate ring of a toric variety”, J. Algebraic Geom., 4 (1995), 17–50 | MR | Zbl

[11] I. Dolgachev, “Weighted projective varieties”, Group Actions and Vector Fields, Lecture Notes in Math., 956, Springer-Verlag, Berlin, 1982, 34–71 | DOI | MR

[12] A. Kasprzyk, “Bounds on fake weighted projective space”, Kodai Math. J., 32:2 (2009), 197–208 | DOI | MR | Zbl

[13] A. A. Borisov, L. A. Borisov, “Osobye toricheskie mnogoobraziya Fano”, Matem. sb., 183:2 (1992), 134–141 | MR | Zbl

[14] A. R. Iano-Fletcher, “Working with weighted complete intersections”, Explicit Birational Geometry of 3-Folds, 281, London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 2000, 101–173 | MR | Zbl

[15] A. Mavlyutov, “Cohomology of complete intersections in toric varieties”, Pacific J. Math., 191:1 (1999), 133–144 | DOI | MR | Zbl

[16] A. Dimca, S. Dimiev, “On analytic coverings of weighted projective spaces”, Bull. London Math. Soc., 17:3 (1985), 234–238 | DOI | MR | Zbl

[17] V. Iskovskikh, Yu. Prokhorov, “Fano varieties”, Algebraic geometry, V, Encyclopaedia Math. Sci., 47, Springer, Berlin, 1999 | MR | Zbl