Blow Up Sequences and the Module of nth Order Differentials

1976 ◽  
Vol 28 (6) ◽  
pp. 1289-1301 ◽  
Author(s):  
William C. Brown
Keyword(s):  
Singular Point ◽  
Local Ring ◽  
Algebraic Curve ◽  
Blow Up ◽  
Quotient Ring ◽  
Integral Closure ◽  
Closed Field ◽  
Quotient Field ◽  
Algebraically Closed Field

Let C denote an irreducible, algebraic curve defined over an algebraically closed field k. Let ? be a singular point of C. We shall employ the following notation throughout the rest of this paper: R will denote the local ring at P, K the quotient field of the integral closure of R in K, A the completion of R with respect to its radical topology, and Ā the integral closure of A in its total quotient ring.

Overrings of Bezout Domains

1973 ◽  
Vol 16 (4) ◽  
pp. 475-477 ◽  
Author(s):  
Raymond A. Beauregard
Keyword(s):  
Local Ring ◽  
Left Ideal ◽  
Quotient Ring ◽  
Valuation Domain ◽  
Quotient Field ◽  
Bezout Domain ◽  
Ideal Domain ◽  
Chain Conditions ◽  
Bezout Domains

In [2] Brungs shows that every ring T between a principal (right and left) ideal domain R and its quotient field is a quotient ring of R. In this note we obtain similar results without assuming the ascending chain conditions. For a (right and left) Bezout domain R we show that T is a quotient ring of R which is again a Bezout domain; furthermore Tis a valuation domain if and only if T is a local ring.

scholarly journals A refined derived Torelli theorem for Enriques surfaces

2020 ◽  
Author(s):  
Chunyi Li ◽  
Howard Nuer ◽  
Paolo Stellari ◽  
Xiaolei Zhao
Keyword(s):  
Simple Proof ◽  
Blow Up ◽  
Derived Categories ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Enriques Surfaces ◽  
Torelli Theorem

AbstractWe prove that two general Enriques surfaces defined over an algebraically closed field of characteristic different from 2 are isomorphic if their Kuznetsov components are equivalent. We apply the same techniques to give a new simple proof of a conjecture by Ingalls and Kuznetsov relating the derived categories of the blow-up of general Artin–Mumford quartic double solids and of the associated Enriques surfaces.

scholarly journals Isomorphisms between complements of projective plane curves

2019 ◽  
Vol Volume 3 ◽  
Author(s):  
Mattias Hemmig
Keyword(s):  
Singular Point ◽  
Projective Plane ◽  
Plane Curves ◽  
Complex Numbers ◽  
Closed Field ◽  
Algebraically Closed Field

In this article, we study isomorphisms between complements of irreducible curves in the projective plane $\mathbb{P}^2$, over an arbitrary algebraically closed field. Of particular interest are rational unicuspidal curves. We prove that if there exists a line that intersects a unicuspidal curve $C \subset \mathbb{P}^2$ only in its singular point, then any other curve whose complement is isomorphic to $\mathbb{P}^2 \setminus C$ must be projectively equivalent to $C$. This generalizes a result of H. Yoshihara who proved this result over the complex numbers. Moreover, we study properties of multiplicity sequences of irreducible curves that imply that any isomorphism between the complements of these curves extends to an automorphism of $\mathbb{P}^2$. Using these results, we show that two irreducible curves of degree $\leq 7$ have isomorphic complements if and only if they are projectively equivalent. Finally, we describe new examples of irreducible projectively non-equivalent curves of degree $8$ that have isomorphic complements.

The Decomposition of the Module of n-th Order Differentials in Arbitrary Characteristic

1978 ◽  
Vol 30 (03) ◽  
pp. 512-517 ◽  
Author(s):  
Klaus G. Fischer
Keyword(s):  
Local Ring ◽  
Valuation Ring ◽  
Algebraic Curve ◽  
Residue Field ◽  
Integral Closure ◽  
Arbitrary Characteristic ◽  
Uniformizing Parameter

Throughout this paper, it is assumed that A is the complete, equicharacteristic, local ring of an algebraic curve at a one-branch singularity whose residue field is algebraically closed and contained in A. Hence, the domain A is dominated by only one valuation ring in its quotient held F, and if t is a uniformizing parameter, then the integral closure of A in F, denoted by Ā, is [[t]].

E∞-Structures on l-Adic Cohomology

2019 ◽  
pp. 129-203
Author(s):  
Dennis Gaitsgory ◽  
Jacob Lurie
Keyword(s):  
Algebraic Geometry ◽  
Algebraic Curve ◽  
Group Scheme ◽  
Affine Group ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Smooth Affine

For applications to Weil's conjecture, a version of (3.1) is formulated in the setting of algebraic geometry, where M is replaced by an algebraic curve X (defined over an algebraically closed field k) and E by the classifying stack BG of a smooth affine group scheme over X. This chapter lays the groundwork by constructing an analogue of the functor B.

scholarly journals Stable vector bundles on an algebraic surface

1975 ◽  
Vol 58 ◽  
pp. 25-68 ◽  
Author(s):  
Masaki Maruyama
Keyword(s):  
Vector Bundle ◽  
Chern Class ◽  
Algebraic Curve ◽  
Algebraic Surface ◽  
Vector Bundles ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Stable Vector Bundles ◽  
First Chern Class

Let X be a non-singular projective algebraic curve over an algebraically closed field k. D. Mumford introduced the notion of stable vector bundles on X as follows;DEFINITION ([7]). A vector bundle E on X is stable if and only if for any non-trivial quotient bundle F of E,where deg ( • ) denotes the degree of the first Chern class of a vector bundles and r( • ) denotes the rank of a vector bundle.

scholarly journals Rings of Fractions and Localization

2020 ◽  
Vol 28 (1) ◽  
pp. 79-87
Author(s):  
Yasushige Watase
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Integral Domain ◽  
Zero Divisor ◽  
Quotient Ring ◽  
Formal Proof ◽  
Universal Property ◽  
Quotient Field ◽  
Closed Set ◽  
Ring Of Fractions

SummaryThis article formalized rings of fractions in the Mizar system [3], [4]. A construction of the ring of fractions from an integral domain, namely a quotient field was formalized in [7].This article generalizes a construction of fractions to a ring which is commutative and has zero divisor by means of a multiplicatively closed set, say S, by known manner. Constructed ring of fraction is denoted by S~R instead of S−1R appeared in [1], [6]. As an important example we formalize a ring of fractions by a particular multiplicatively closed set, namely R \ p, where p is a prime ideal of R. The resulted local ring is denoted by Rp. In our Mizar article it is coded by R~p as a synonym.This article contains also the formal proof of a universal property of a ring of fractions, the total-quotient ring, a proof of the equivalence between the total-quotient ring and the quotient field of an integral domain.

scholarly journals On certain isolated normal singularities

1976 ◽  
Vol 61 ◽  
pp. 205-220 ◽  
Author(s):  
Lucian Bădescu
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Arbitrary Characteristic ◽  
Algebraic Scheme

In the following we shall fix an algebraically closed fieldKof arbitrary characteristic. The term variety will mean an algebraic scheme overKwhich is integral. In general we shall use the notations and the terminology of Éléments de Géométrie Algébrique of A. Grothendieck and J. Dieudonné. For instance, ifZis a variety andx ∈ Zis a point, thenOZ, xmeans the local ring ofZatxandmx– the maximal ideal of0Z, x.

scholarly journals The strong simple connectedness of tame algebras with separating almost cyclic coherent Auslander–Reiten components

2021 ◽  
Author(s):  
Piotr Malicki
Keyword(s):  
Closed Field ◽  
Algebraically Closed Field ◽  
Main Application ◽  
Finite Dimensional ◽  
Simple Connectedness ◽  
Tame Algebras

AbstractWe study the strong simple connectedness of finite-dimensional tame algebras over an algebraically closed field, for which the Auslander–Reiten quiver admits a separating family of almost cyclic coherent components. As the main application we describe all analytically rigid algebras in this class.

scholarly journals On Unramified Separable Abelian p-Extensions of Function Fields I

1959 ◽  
Vol 14 ◽  
pp. 223-234 ◽  
Author(s):  
Hisasi Morikawa
Keyword(s):  
Galois Group ◽  
Function Field ◽  
Function Fields ◽  
Abelian Extension ◽  
Closed Field ◽  
Jacobian Variety ◽  
Algebraically Closed Field

Let k be an algebraically closed field of characteristic p>0. Let K/k be a function field of one variable and L/K be an unramified separable abelian extension of degree pr over K. The galois automorphisms ε1, …, εpr of L/K are naturally extended to automorphisms η(ε1), … , η(εpr) of the jacobian variety JL of L/k. If we take a svstem of p-adic coordinates on JL, we get a representation {Mp(η(εv))} of the galois group G(L/K) of L/K over p-adic integers.

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