Blowups with log canonical singularities

Sankaran, Gregory¹ ; Santos, Francisco²

¹ Department of Mathematical Sciences, University of Bath, Bath, United Kingdom
² Departamento de Matemáticas, Estadística y Computación, Facultad de Ciencias, Universidad de Cantabria, Santander, Spain

Geometry & topology, Tome 25 (2021) no. 4, pp. 2145-2166.

Résumé

We show that the minimum weight of a weighted blowup of ${\mathbb{𝔸}}^d$ with $𝜀$–log canonical singularities is bounded by a constant depending only on $𝜀$ and $d$. This was conjectured by Birkar.

Using the recent classification of $4$–dimensional empty simplices by Iglesias-Valiño and Santos, we work out an explicit bound for blowups of ${\mathbb{𝔸}}^4$ with terminal singularities: the smallest weight is always at most $32$, and at most $6$ in all but finitely many cases.

Détail

DOI : 10.2140/gt.2021.25.2145

Classification : 14B05, 14E99, 14M25, 52B20
Keywords: binational geometry, log canonical singularities, blowups, lattice simplices

Bibliographie

[1] R E Anno, Four-dimensional terminal Gorenstein quotient singularities, Mat. Zametki 73 (2003) 813

[2] M Barile, D Bernardi, A Borisov, J M Kantor, On empty lattice simplices in dimension 4, Proc. Amer. Math. Soc. 139 (2011) 4247 | DOI

[3] C Birkar, Anti-pluricanonical systems on Fano varieties, Ann. of Math. 190 (2019) 345 | DOI

[4] M Blanco, C Haase, J Hofmann, F Santos, The finiteness threshold width of lattice polytopes, Trans. Amer. Math. Soc. Ser. B 8 (2021) 399 | DOI

[5] J W Bober, Factorial ratios, hypergeometric series, and a family of step functions, J. Lond. Math. Soc. 79 (2009) 422 | DOI

[6] A Borisov, On classification of toric singularities, J. Math. Sci. (New York) 94 (1999) 1111 | DOI

[7] Y Chen, On singularities of threefold weighted blowups, J. Math. Soc. Japan (2021) 1 | DOI

[8] C A J Hurkens, Blowing up convex sets in the plane, Linear Algebra Appl. 134 (1990) 121 | DOI

[9] Ó Iglesias-Valiño, F Santos, The complete classification of empty lattice 4–simplices, Rev. Mat. Iberoam. (2021) | DOI

[10] M Kawakita, Divisorial contractions in dimension three which contract divisors to smooth points, Invent. Math. 145 (2001) 105 | DOI

[11] J Lawrence, Finite unions of closed subgroups of the n–dimensional torus, from: "Applied geometry and discrete mathematics" (editors P Gritzmann, B Sturmfels), DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 4, Amer. Math. Soc. (1991) 433 | DOI

[12] S Mori, D R Morrison, I Morrison, On four-dimensional terminal quotient singularities, Math. Comp. 51 (1988) 769 | DOI

[13] D R Morrison, G Stevens, Terminal quotient singularities in dimensions three and four, Proc. Amer. Math. Soc. 90 (1984) 15 | DOI

[14] T Oda, Convex bodies and algebraic geometry, 15, Springer (1988)

[15] G K Sankaran, Stable quintuples and terminal quotient singularities, Math. Proc. Cambridge Philos. Soc. 107 (1990) 91 | DOI

[16] G K White, Lattice tetrahedra, Canad. J. Math. 16 (1964) 389 | DOI