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Kenji Matsuki

Publications and source records attributed to Kenji Matsuki.

14 recordsLinked to original sources
arXiv2023

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

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.

math.AG↗
arXiv2015

A new strategy for resolution of singularities in the monomial case in positive characteristic

According to our approach for resolution of singularities in positive characteristic (called the Idealistic Filtration Program, alias the I.F.P. for short) the algorithm is devided into the following two steps: Step 1. Reduction of the general case to the monomial case. Step 2. Solution in the monomial case. While we have established Step 1 in arbitrary dimension, Step 2 becomes very subtle and difficult in positive characteristic. This is in clear contrast to the classical setting in characteristic zero, where the solution in the monomial case is quite easy. In dimension 3, we provided an invariant in the previous paper, inspired by the work of Benito-Villamayor, which establishes Step 2. In this paper, we propose a new strategy to approach Step 2, and provide a different invariant in dimension 3 based upon this strategy. The new invariant increases from time to time (the well-known Moh-Hauser jumping phoenomena), while it is then shown to eventually decrease. (The analysis of the jumping phoenomena and eventual decrease is done in the monomial case in our setting, while the classical analysis by Moh or Hauser is done in a different setting without any reference to the monomial case. Therefore, even though we owe most of the ideas to Moh and Hauser, our argument is carried out logically independent of their papers.) Since the old invariant in our previous paper strictly decreases after each transformation, this may look like a step backward rather than forward. However, the construction of the new invariant is more faithful to the original philosophy of Villamayor, and we believe that the new strategy has a better fighting chance in higher dimensions.

math.AG↗
arXiv2012

Resolution of singularities of an idealistic filtration in dimension 3 after Benito-Villamayor

We establish an algorithm for resolution of singularities of an idealistic filtration in dimension 3 (at the local level) in positive characteristic, incorporating the method recently developed by Benito-Villamayor into our framework. Although (a global version of) our algorithm only implies embedded resolution of surfaces in the smooth ambient space of dimension 3, a classical result known before, we introduce some new invariant which effectively measures how much singularities are improved in the process of our algorithm and which strictly drops after each blow up. This is in contrast to the well-known Abhyankar-Moh pathology of the increase of the residual order under blow up and the phenomenon of the "Kangaroo" points observed by Hauser.

math.AG↗
arXiv2003

A correction to the paper "Log Abundance Theorem for Threefolds"

This is a correction to the afore-mentioned paper in Duke Math. J. vol. 75 (1994), 99-119 by S. Keel, K. Matsuki, and J. McKernan. We completely rewrite Chapter 6 according to the original manuscript of the second author, in order to fix some crucial mistakes pointed out by Dr. Qihong Xie.

math.AG↗
arXiv2002

Kawamata-Viehweg vanishing as Kodaira vanishing for stacks

We associate to a pair $(X,D)$, consisting of a smooth scheme with a divisor $D\in \text{Div}(X)\otimes \mathbb{Q}$ whose support is a divisor with normal crossings, a canonical Deligne--Mumford stack over $X$ on which $D$ becomes integral. We then reinterpret the Kawamata--Viehweg vanishing theorem as Kodaira vanishing for stacks.

math.AG↗
arXiv2002

Notes on the inductive algorithm of resolution of singularities by S. Encinas and O. Villamayor

In this revised version, the mistake of the author confusing the weak transform and strict transform, pointed out by E. Bierstone, is corrected. It gives a self-contained proof of (embedded) resolution of singularities over a field of characteristic zero following the inductive algorithm of Encinas and Villamayor. A chapter explaining the background motivation for the invariants used (based upon a lecture by Villamayor) and another containing some examples demonstrating the subtle points are added.

math.AG↗
arXiv2000

Erratum to the paper "A note on the factorization theorem of toric birational maps after Morelli and its toroidal extension"

This is an erratum to math.AG/9803126, Tohoku 51 (1999) 489-537. This erratum describes: 1. the failure of the algorithm in [AMR] and [Morelli1] for the strong factorization pointed out by Kalle Karu, 2. the statement of a refined weak factorization theorem for toroidal birational morphisms in [AMR], in the form utilized in [AKMR] for the proof of the weak factorization theorem for general birationla maps, avoiding the use of the above mentioned algorithm for the strong factorization, and 3. a list of corrections for a few other mistakes in [AMR], mostly pointed out by Laurent Bonavero.

math.AG↗
arXiv2000

Torification and Factorization of Birational Maps

Building on the work of the fourth author in math.AG/9904074, we prove the weak factorization conjecture for birational maps in characteristic zero: a birational map between complete nonsingular varieties over an algebraically closed field K of characteristic zero is a composite of blowings up and blowings down with smooth centers. Such a factorization exists which is functorial with respect to absolute isomorphisms, and compatible with a normal crossings divisor. The same holds for algebraic and analytic spaces. Another proof of the main theorem by the fourth author appeared in math.AG/9904076.

math.AG↗
arXiv2000

Lectures on Factorization of Birational Maps

This is an expanded version of the notes for the lectures given by the author at RIMS in the summer of 1999 to give a detailed account of the proof for the (weak) factorization theorem of birational maps by Abramovich-Karu-Matsuki-Włodarczyk.

math.AG↗
arXiv1998

Uniformity of stably integral points on principally polarized abelian surfaces

We prove, assuming that the conjecture of Lang and Vojta holds true, that there is a uniform bound on the number of stably integral points in the complement of the theta divisor on a principally polarized abelian surface defined over a number field. This gives a uniform version, in the spirit of a result of Caporaso-Harris-Mazur, of an unconditional theorem of Faltings. We utilize recent results of Alexeev and Nakamura on complete moduli for quasi-abelian varieties. We expect that a thorough understanding of current work of Alexeev should give a more general result for abelian varieties of an arbitrary dimension with a polarizing divisor of an arbitrary degree - a proposed approach for such a generalization is given at the end of the paper.

math.AG↗
arXiv1995

Log Sarkisov Program

The purpose of this paper is two-fold. The first is to give a tutorial introduction to the Sarkisov program, a 3-dimensional generalization of Castelnuovo-N\"other Theorem ``untwisting" birational maps between Mori fiber spaces, which was recently established by Corti after Sarkisov and Reid. The second is an attempt to give a logarithmic generalization of the program, introducing the notion of the Sarkisov relation. We give a simple description of the structure of the program and clarify the mechanism of termination connected to the boundedness conjecture of (log) Q-Fano n-folds by Borisov.

alg-geom↗