scholarly journals Seminormal graded rings and weakly normal projective varieties

1985 ◽  
Vol 8 (2) ◽  
pp. 231-240 ◽  
Author(s):  
John V. Leahy ◽  
Marie A. Vitulli
Keyword(s):  
Graded Rings ◽  
Projective Varieties ◽  
Normal Variety ◽  
Blowing Up ◽  
Weak Normality

This paper is concerned with the seminormality of reduced graded rings and the weak normality of projective varieties. One motivation for this investigation is the study of the procedure of blowing up a non-weakly normal variety along its conductor ideal.

ON THE COHEN–MACAULAYNESS OF SOME GRADED RINGS

2008 ◽  
Vol 07 (01) ◽  
pp. 109-128
Author(s):  
D. P. PATIL ◽  
G. TAMONE
Keyword(s):  
Numerical Semigroup ◽  
Sufficient Conditions ◽  
Local Rings ◽  
Graded Rings ◽  
Semigroup Rings ◽  
Associated Graded Ring ◽  
Blowing Up ◽  
The Difference ◽  
Necessary And Sufficient ◽  
Numerical Semigroup Rings

Let (R,𝔪) be a 1-dimensional Cohen–Macaulay local ring of multiplicity e and embedding dimension ν ≥ 2. Let B denote the blowing-up of R along 𝔪 and let I be the conductor of R in B. Let x ∈ 𝔪 be a superficial element in 𝔪 of degree 1 and [Formula: see text], [Formula: see text]. We assume that the length [Formula: see text]. This class of local rings contains the class of 1-dimensional Gorenstein local rings (see 1.5). In Sec. 1, we prove that (see 1.6) if the associated graded ring G = gr 𝔪(R) is Cohen–Macaulay, then I ⊆ 𝔪s + xR, where s is the degree of the h-polynomial h R of R. In Sec. 2, we give necessary and sufficient conditions (see Corollaries 2.4, 2.5, 2.9 and Theorem 2.11) for the Cohen–Macaulayness of G. These conditions are numerical conditions on the h-polynomial h R, particularly on its coefficients and the degree in comparison with the difference e - ν. In Sec. 3, we give some conditions (see Propositions 3.2, 3.3 and Corollary 3.4) for the Gorensteinness of G. In Sec. 4, we give a characterization (see Proposition 4.3) of numerical semigroup rings which satisfy the condition [Formula: see text].

On blowing up conductor ideals

1978 ◽  
Vol 83 (3) ◽  
pp. 445-450 ◽  
Author(s):  
P. M. H. Wilson
Keyword(s):  
Algebraic Geometry ◽  
Blow Up ◽  
Normal Variety ◽  
Blowing Up

In this paper we investigate the procedure of blowing up a non-normal variety V in its conductor ideal (denned in Section 1). If V is a hypersurface then corresponds to the subadjunction conditions of Italian algebraic geometry (see (4), section 15), and so we would expect the blow up of V in , denoted , to be closely connected with the normalization Ṽ.

A generalisation of the Abhyankar Jung theorem to associated graded rings of valuations

2015 ◽  
Vol 160 (2) ◽  
pp. 233-255 ◽  
Author(s):  
STEVEN DALE CUTKOSKY
Keyword(s):  
Characteristic Zero ◽  
Positive Characteristic ◽  
Natural Extension ◽  
Graded Rings ◽  
Blowing Up ◽  
Associated Graded Rings

AbstractSuppose thatR→Sis an extension of local domains andν* is a valuation dominatingS. We consider the natural extension of associated graded rings along the valuation grν*(R) → grν*(S). We give examples showing that in general, this extension does not share good properties of the extensionR→S, but after enough blow ups above the valuations, good properties of the extensionR→Sare reflected in the extension of associated graded rings. Stable properties of this extension (after blowing up) are much better in characteristic zero than in positive characteristic. Our main result is a generalisation of the Abhyankar–Jung theorem which holds for extensions of associated graded rings along the valuation, after enough blowing up.

On 2-dimensional local rings with Artin's approximation property

1983 ◽  
Vol 94 (1) ◽  
pp. 35-52
Author(s):  
M. L. Brown
Keyword(s):  
Characteristic Zero ◽  
Approximation Property ◽  
Local Rings ◽  
Projective Varieties ◽  
Blowing Up

AbstractExtending results of Popescu and Brown, the main result of this paper is that excellent henselian R1 and S1 2-dimensional local rings, at least in characteristic zero, have the approximation property of M. Artin.Most of the paper consists of an extension of Néron's desingularization to rings which are R1 and S1; such a theorem was previously known for factorial domains. The main theorem is then deduced from this desingularization theorem using a theorem of Elkik.Because of cohomological obstructions, the desingularization theorem is proved only for quasi-projective varieties. In the previously known case for factorial domains, these obstructions are always zero and the desingularization can be obtained by blowing up subschemes. The more general desingularization of this paper is obtained by blowing up locally free sheaves instead, the obstructions being zero for this case.

On the Curves Associated to Certain Rings of Automorphic Forms

2001 ◽  
Vol 53 (1) ◽  
pp. 98-121 ◽  
Author(s):  
Kamal Khuri-Makdisi
Keyword(s):  
Ad Hoc ◽  
Automorphic Forms ◽  
Complex Multiplication ◽  
Graded Rings ◽  
Projective Varieties ◽  
Hecke Operator ◽  
Complex Tori ◽  
Hecke Correspondences ◽  
Cm Points ◽  
Projective Lines

AbstractIn a 1987 paper, Gross introduced certain curves associated to a definite quaternion algebra B over Q; he then proved an analog of his result with Zagier for these curves. In Gross’ paper, the curves were defined in a somewhat ad hoc manner. In this article, we present an interpretation of these curves as projective varieties arising from graded rings of automorphic forms on B×, analogously to the construction in the Satake compactification. To define such graded rings, one needs to introduce a “multiplication” of automorphic forms that arises from the representation ring of B×. The resulting curves are unions of projective lines equipped with a collection of Hecke correspondences. They parametrize two-dimensional complex tori with quaternionic multiplication. In general, these complex tori are not abelian varieties; they are algebraic precisely when they correspond to CM points on these curves, and are thus isogenous to a product E × E, where E is an elliptic curve with complex multiplication. For these CM points one can make a relation between the action of the p-th Hecke operator and Frobenius at p, similar to the well-known congruence relation of Eichler and Shimura.

scholarly journals F-THRESHOLDS OF GRADED RINGS

2016 ◽  
Vol 229 ◽  
pp. 141-168 ◽  
Author(s):  
ALESSANDRO DE STEFANI ◽  
LUIS NÚÑEZ-BETANCOURT
Keyword(s):  
Characteristic Zero ◽  
Projective Dimension ◽  
Graded Rings ◽  
Projective Varieties ◽  
Regular Sequences ◽  
Serre’S Condition ◽  
Regular Rings

The $a$-invariant, the $F$-pure threshold, and the diagonal $F$-threshold are three important invariants of a graded $K$-algebra. Hirose, Watanabe, and Yoshida have conjectured relations among these invariants for strongly $F$-regular rings. In this article, we prove that these relations hold only assuming that the algebra is $F$-pure. In addition, we present an interpretation of the $a$-invariant for $F$-pure Gorenstein graded $K$-algebras in terms of regular sequences that preserve $F$-purity. This result is in the spirit of Bertini theorems for projective varieties. Moreover, we show connections with projective dimension, Castelnuovo–Mumford regularity, and Serre’s condition $S_{k}$. We also present analogous results and questions in characteristic zero.

scholarly journals Some remarks concerning Demazure’s construction of normal graded rings

1981 ◽  
Vol 83 ◽  
pp. 203-211 ◽  
Author(s):  
Keiichi Watanabe
Keyword(s):  
Finite Type ◽  
Homogeneous Element ◽  
Rational Coefficient ◽  
Graded Rings ◽  
Quotient Field ◽  
Graded Ring ◽  
Projective Varieties ◽  
Definition Of

In [1], Demazure showed a new way of constructing normal graded rings using the concept of “rational coefficient Weil divisors” of normal projective varieties and he showed, among other things, the followingTHEOREM ([1], 3.5). If R = ⊕n ≥ 0Rn is a normal graded ring of finite type over a field k and if T is a homogeneous element of degree 1 in the quotient field of R, then there exists unique divisor D ∈ Div (X, Q) (X = Proj (R)), such that for every n ≧ 0.(See (1.1) for the definition of

Cohomological Conditions on Endomorphisms of Projective Varieties

2017 ◽  
Vol 145 (3) ◽  
pp. 449-468 ◽  
Author(s):  
Holly Krieger ◽  
Paul Reschke
Keyword(s):  
Projective Varieties
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