arXiv subjects
Publications and source records attributed to Kei-ichi Watanabe.
In this paper, from an arbitrary smooth projective curve of genus at least two, we construct a non-Gorenstein Cohen-Macaulay normal domain and a nonfree totally reflexive module over it.
We introduce and study invariants of singularities in positive characteristic called F-thresholds. They give an analogue of the jumping coefficients of multiplier ideals in characteristic zero. We discuss the connection between the invariants of an ideal in characteristic zero and the invariants of the different reduction mod p of this ideal. Our main point is that this relation depends on arithmetic properties of p. We also describe a new connection between invariants mod p and the roots of the Bernstein-Sato polynomial.
Using the Frobenius map, we introduce a new invariant for a pair $(R,\a)$ of a ring $R$ and an ideal $\a \subset R$, which we call the F-pure threshold $\mathrm{c}(\a)$ of $\a$, and study its properties. We see that the F-pure threshold characterizes several ring theoretic properties. By virtue of Hara and Yoshida's result, the F-pure threshold $\mathrm{c}(\a)$ in characteristic zero corresponds to the log canonical threshold $\mathrm{lc}(\a)$ which is an important invariant in birational geometry. Using the F-pure threshold, we prove some ring theoretic properties of three-dimensional terminal singularities of characteristic zero. Also, in fixed prime characteristic, we establish several properties of F-pure threshold similar to those of the log canonical threshold with quite simple proofs.
The total coordinate ring TC(X) of a normal variety is a generalization of the ring introduced and studied by Cox in connection with a toric variety. Consider a normal projective variety X with divisor class group Cl(X), and let us assume that it is a finitely generated free abelian group. We define the total coordinate ring of X to be TC(X) = oplus_{D} H^0 (X, O_X (D)), where the sum as above is taken over all Weil divisors of X contained in a fixed complete system of representatives of Cl(X). We prove that for any normal projective variety X, TC(X) is a UFD, this is a corollary of a more general theorem that is proved in the paper. (Berchtold and Haussen proved the unique factorization for a smooth variety independently.) We also prove that for X, the blow up of P^2 along a finite number of collinear points, TC(X) is Noetherian. We also give an example that TC(X) is not Noetherian but oplus_n H^0 (X, O(nD)) is Noetherian for any Weil divisor D.
In this paper, we investigate a lower bound (say $s_{HK}(p,d)$) on Hilbert-Kunz multiplicities for non-regular unmixed local rings of Krull dimension $d$ with characteristic $p>0$. Especially, we focus three-dimensional local rings. In fact, as a main result, we will prove that $s_{HK}(p,3) = 4/3$ and that a three-dimensional complete local ring of Hilbert-Kunz multiplicity 4/3 is isomorphic to the non-degnerate quadric hyperplanes $k[[X,Y,Z,W]]/(X^2+Y^2+Z^2+W^2)$ under mild conditions. Furthermore, we pose a generalization of the main theorem to the case of $\dim A \ge 4$ as a conjecture, and show that it is also true in case of $\dim A = 4$ using the similar method as in the proof of the main theorem.
Demailly, Ein and Lazarsfeld \cite{DEL} proved the subadditivity theorem for multiplier ideals, which states the multiplier ideal of the product of ideals is contained in the product of the individual multiplier ideals, on non-singular varieties. We prove that, in two-dimensional case, the subadditivity theorem holds on log-terminal singularities. However, in higher dimensional case, we have several counter-examples. We consider the subadditivity theorem for monomial ideals on toric rings, and construct a counter-example on a three-dimensional toric ring.
In this paper, we ask the following question: what is the minimal value of the difference $e_{HK}(I) - e_{HK}(I')$ for ideals $I' \supseteq I$ with $l_A(I'/I) =1$? In order to answer to this question, we define the notion of minimal Hilbert-Kunz multiplicity for strongly F-regular rings. Moreover, we calculate this invariant for quotient singularities and for the coordinate ring of the Segre embedding: $P^{r-1} \times P^{s-1} \hookrightarrow P^{rs-1}$, respectively.
Let $(A,\frak m)$ be an excellent normal local ring with algebraically closed residue class field. Given integrally closed $\frak m$-primary ideals $I\supset J$, we show that there is a composition series between $I$ and $J$, by integrally closed ideals only. Also we show that any given integrally closed $\fm$-primary ideal $I$, the family of integrally closed ideals $J\subset I, l_A(I/J)=1$ forms an algebraic variety with dimension $\dim A -1$.
In this paper, we investigate the relationship of F-regular (resp. F-pure) rings and log terminal (resp. log canonical) singularities. Also, we extend the notions of F-regularity and F-purity to "F-singularities of pairs." The notions of F-regular and F-pure rings in characteristic $p>0$ are characterized by a splitting of the Frobenius map, and define some classes of rings having "mild" singularities. On the other hand, there are notions of log terminal and log canonical singularities defined via resolutions of singularities in characteristic zero. These are defined also for pairs of a normal variety and a $\Bbb Q$-divisor $Δ$ on it, and play an important role in birational algebraic geometry. As an analog of these singularities of pairs in characteristic zero, we define the notions of "F-singularities of pairs," namely strong F-regularity, divisorial F-regularity and F-purity for a pair $(A,Δ)$ of a normal ring $A$ of characteristic $p > 0$ and an effective $\Bbb Q$-divisor $Δ$ on $Y = Spec A$. The main theorem of this paper asserts that, if $K_Y + Δ$ is $\Bbb Q$-Cartier, then the above three variants of F-singularitiesof pairs imply KLT, PLT and LC properties, respectively. We also prove some results for F-singularities of pairs which are analoguous to singularities of pairs in characteristic zero.