627 research outputs found
Numerical equivalence defined on Chow groups of Noetherian local rings
In the present paper, we define a notion of numerical equivalence on Chow
groups or Grothendieck groups of Noetherian local rings, which is an analogue
of that on smooth projective varieties. Under a mild condition, it is proved
that the Chow group modulo numerical equivalence is a finite dimensional -vector space, as in the case of smooth projective varieties. Numerical
equivalence on local rings is deeply related to that on smooth projective
varieties. For example, if Grothendieck's standard conjectures are true, then a
vanishing of Chow group (of local rings) modulo numerical equivalence can be
proven. Using the theory of numerical equivalence, the notion of numerically
Roberts rings is defined. It is proved that a Cohen-Macaulay local ring of
positive characteristic is a numerically Roberts ring if and only if the
Hilbert-Kunz multiplicity of a maximal primary ideal of finite projective
dimension is always equal to its colength. Numerically Roberts rings satisfy
the vanishing property of intersection multiplicities. We shall prove another
special case of the vanishing of intersection multiplicities using a vanishing
of localized Chern characters.Comment: final version, 45 pages, to appear in Invent. Mat
The singular Riemann-Roch theorem and Hilbert-Kunz functions
In the paper, by the singular Riemann-Roch theorem, it is proved that the
class of the e-th Frobenius power can be described using the class of the
canonical module for a normal local ring of positive characteristic. As a
corollary, we prove that the coefficient of the second term of the Hilbert-Kunz
function of a finitely generated A-module M vanishes if A is a Q-Gorenstein
ring and M is of finite projective dimension. For a normal algebraic variety X
over a perfect field of positive characteristic, it is proved that the first
Chern class of the direct image of the structure sheaf via e-th Frobenius power
can be described using the canonical divisor of X.Comment: 12 pages. to appear in J. Algebr
On finite generation of symbolic Rees rings of space monomial curves and existence of negative curves
In this paper, we shall study finite generation of symbolic Rees rings of the
defining ideal of the space monomial curves for pairwise
coprime integers , , such that . If such a ring
is not finitely generated over a base field, then it is a counterexample to the
Hilbert's fourteenth problem. Finite generation of such rings is deeply related
to existence of negative curves on certain normal projective surfaces. We study
a sufficient condition (Definition 3.6) for existence of a negative curve.
Using it, we prove that, in the case of , a negative curve
exists. Using a computer, we shall show that there exist examples in which this
sufficient condition is not satisfied.Comment: In the previous version, there was a serious mistake in the last
sectio
The canonical module of a Cox ring
In this paper, we shall describe the graded canonical module of a Noetherian
multi-section ring of a normal projective variety. In particular, in the case
of the Cox ring, we prove that the graded canonical module is a graded free
module of rank one with the shift of degree . We shall give two kinds of
proofs. The first one utilizes the equivariant twisted inverse functor
developed by the first author. The second proof is down-to-earth, that avoids
the twisted inverse functor.Comment: 19 pages, corrected minor errors and updated the reference
Asymptotic regularity of powers of ideals of points in a weighted projective plane
In this paper we study the asymptotic behavior of the regularity of symbolic
powers of ideals of points in a weighted projective plane. By a result of
Cutkosky, Ein and Lazarsfeld, regularity of such powers behaves asymptotically
like a linear function. We study the difference between regularity of such
powers and this linear function. Under some conditions, we prove that this
difference is bounded, or eventually periodic. As a corollary we show that, if
there exists a negative curve, then the regularity of symbolic powers of a
monomial space curve is eventually a periodic linear function. We give a
criterion for the validity of Nagata's conjecture in terms of the lack of
existence of negative curves.Comment: 16 page
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