### Abstract

Let (X, Δ) be an n-dimensional ϵ-klt log ℚ-Fano pair. We give an upper bound for the volume Vol(X, Δ) = (-(K_{X} + Δ))^{n} when n = 2 , or n = 3 and X is ℚ-factorial of ρ(X)=1 . This bound is essentially sharp for . The main idea is to analyze the covering families of tigers constructed in J.Â M^{c}Kernan (Boundedness of log terminal fano pairs of bounded index, preprint, 2002, arXiv:0205214). Existence of an upper bound for volumes is related to the Borisov-Alexeev-Borisov Conjecture, which asserts boundedness of the set of ϵ-klt log ℚ-Fano varieties of a given dimension.

Original language | English |
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Pages (from-to) | 37-73 |

Number of pages | 37 |

Journal | Nagoya Mathematical Journal |

Volume | 224 |

DOIs | |

Publication status | Published - 2016 Dec 1 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

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*Nagoya Mathematical Journal*, vol. 224, pp. 37-73. https://doi.org/10.1017/nmj.2016.21

**Bounding volumes of singular fano threefolds.** / Lai, Ching Jui.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Bounding volumes of singular fano threefolds

AU - Lai, Ching Jui

PY - 2016/12/1

Y1 - 2016/12/1

N2 - Let (X, Δ) be an n-dimensional ϵ-klt log ℚ-Fano pair. We give an upper bound for the volume Vol(X, Δ) = (-(KX + Δ))n when n = 2 , or n = 3 and X is ℚ-factorial of ρ(X)=1 . This bound is essentially sharp for . The main idea is to analyze the covering families of tigers constructed in J.Â McKernan (Boundedness of log terminal fano pairs of bounded index, preprint, 2002, arXiv:0205214). Existence of an upper bound for volumes is related to the Borisov-Alexeev-Borisov Conjecture, which asserts boundedness of the set of ϵ-klt log ℚ-Fano varieties of a given dimension.

AB - Let (X, Δ) be an n-dimensional ϵ-klt log ℚ-Fano pair. We give an upper bound for the volume Vol(X, Δ) = (-(KX + Δ))n when n = 2 , or n = 3 and X is ℚ-factorial of ρ(X)=1 . This bound is essentially sharp for . The main idea is to analyze the covering families of tigers constructed in J.Â McKernan (Boundedness of log terminal fano pairs of bounded index, preprint, 2002, arXiv:0205214). Existence of an upper bound for volumes is related to the Borisov-Alexeev-Borisov Conjecture, which asserts boundedness of the set of ϵ-klt log ℚ-Fano varieties of a given dimension.

UR - http://www.scopus.com/inward/record.url?scp=85019561201&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85019561201&partnerID=8YFLogxK

U2 - 10.1017/nmj.2016.21

DO - 10.1017/nmj.2016.21

M3 - Article

AN - SCOPUS:85019561201

VL - 224

SP - 37

EP - 73

JO - Nagoya Mathematical Journal

JF - Nagoya Mathematical Journal

SN - 0027-7630

ER -