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Addendum/Erratum to the paper "Weyl groups and Birational transformations among minimal models"

Kenji Matsuki

Abstract

We present an addendum/erratum to the paper "Weyl Groups and Birational Transformations among Minimal Models" written by the author and published in 1995, adding the analysis of the "88-th" deformation type of a smooth Fano 3-fold with $B_2 = 4$ denoted as $n^o\ 13$, which was missing from the original classification table by Mori-Mukai and added later to the list of smooth Fano 3-folds with $B_2 \geq 2$, while correcting the mistake pointed out by Eric Jovinelly (also noticed earlier by Kento Fujita). We also correct other typos and miscalculations, clarifying some points of ambiguity. The original paper is an attempt to generalize the result of Y. Manin associating some Weyl groups to Del Pezzo surfaces to the one associating certain Weyl groups to Fano 3-folds.

Addendum/Erratum to the paper "Weyl groups and Birational transformations among minimal models"

Abstract

We present an addendum/erratum to the paper "Weyl Groups and Birational Transformations among Minimal Models" written by the author and published in 1995, adding the analysis of the "88-th" deformation type of a smooth Fano 3-fold with denoted as , which was missing from the original classification table by Mori-Mukai and added later to the list of smooth Fano 3-folds with , while correcting the mistake pointed out by Eric Jovinelly (also noticed earlier by Kento Fujita). We also correct other typos and miscalculations, clarifying some points of ambiguity. The original paper is an attempt to generalize the result of Y. Manin associating some Weyl groups to Del Pezzo surfaces to the one associating certain Weyl groups to Fano 3-folds.
Paper Structure (13 sections, 1 theorem, 64 equations)

This paper contains 13 sections, 1 theorem, 64 equations.

Table of Contents

  1. Outline of the addendum/erratum
  2. Claim III-3-2 revisited
  3. Claim III-3-2
  4. Proof of Claim III-3-2
  5. Analysis of the Fano 3-fold denoted as $n^o 1$ with $B_2 \geq 5$ in Table 5
  6. Analysis of the Fano 3-fold denoted as $n^o\ 13$ with $B_2 = 4$ in Table 4
  7. Clarification of the statements about the types of our 4-fold flops
  8. Origin of the naming of the types of flops
  9. Theorem III-2-1
  10. Clarification of the meaning of the word "get"
  11. Labeling ($F$) in the item (4)
  12. Correction to THEOREM III-2-4
  13. Mistakes found in the tables of the intersection pairings and the corrections

Key Result

Proposition 2.1.1

Let $L = \{l_1, l_2, \ldots, l_m\}$ be a given set of extremal rays on a smooth Fano 3-fold $T$. Take and fix an extremal ray $l_i \in L$. Then the contraction of the extremal ray $l_i$, denoted by $\mathrm{cont}_{l_i}: T \rightarrow U_i$, gives another $\mathbb{Q}$-factorial variety $U_i$. (We note

Theorems & Definitions (1)

  • Proposition 2.1.1