The Conductor of Points Having the Hilbert Function of a Complete Intersection in P2

1992 ◽  
Vol 44 (1) ◽  
pp. 167-179 ◽  
Author(s):  
Amar Sodhi
Keyword(s):  
Complete Intersection ◽  
Hilbert Function ◽  
Coordinate Ring ◽  
Coordinate Rings

AbstractLet A be the coordinate ring of a set of s points in pn(k). After examining what the Hilbert function of A tells us about the conductor of A, we then determine the possible conductors for those coordinate rings which have the Hilbert function of a complete intersection in P2(k).

scholarly journals Maps from Feigin and Odesskii's elliptic algebras to twisted homogeneous coordinate rings

2021 ◽  
Vol 9 ◽  
Author(s):  
Alex Chirvasitu ◽  
Ryo Kanda ◽  
S. Paul Smith
Keyword(s):  
Elliptic Curve ◽  
Symmetric Power ◽  
Canonical Homomorphism ◽  
Coordinate Ring ◽  
Characteristic Variety ◽  
Closed Subvariety ◽  
Coordinate Rings ◽  
Cubic Hypersurfaces ◽  
Elliptic Algebras ◽  
Twisted Homogeneous Coordinate Ring

Abstract The elliptic algebras in the title are connected graded $\mathbb {C}$ -algebras, denoted $Q_{n,k}(E,\tau )$ , depending on a pair of relatively prime integers $n>k\ge 1$ , an elliptic curve E and a point $\tau \in E$ . This paper examines a canonical homomorphism from $Q_{n,k}(E,\tau )$ to the twisted homogeneous coordinate ring $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ on the characteristic variety $X_{n/k}$ for $Q_{n,k}(E,\tau )$ . When $X_{n/k}$ is isomorphic to $E^g$ or the symmetric power $S^gE$ , we show that the homomorphism $Q_{n,k}(E,\tau ) \to B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ is surjective, the relations for $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ are generated in degrees $\le 3$ and the noncommutative scheme $\mathrm {Proj}_{nc}(Q_{n,k}(E,\tau ))$ has a closed subvariety that is isomorphic to $E^g$ or $S^gE$ , respectively. When $X_{n/k}=E^g$ and $\tau =0$ , the results about $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ show that the morphism $\Phi _{|\mathcal {L}_{n/k}|}:E^g \to \mathbb {P}^{n-1}$ embeds $E^g$ as a projectively normal subvariety that is a scheme-theoretic intersection of quadric and cubic hypersurfaces.

Integral closure of some co-ordinate rings

1975 ◽  
Vol 20 (1) ◽  
pp. 115-123
Author(s):  
David J. Smith
Keyword(s):  
Integral Closure ◽  
Principal Result ◽  
Space Curve ◽  
Coordinate Ring ◽  
Unique Factorization ◽  
Algebraic Space ◽  
Space Curves ◽  
Integrally Closed ◽  
Coordinate Rings ◽  
Special Case

In this paper, some methods are developed for obtaining explicitly a basis for the integral closure of a class of coordinate rings of algebraic space curves.The investigation of this problem was motivated by a need for examples of integrally closed rings with specified subrings with a view toward examining questions of unique factorization in them. The principal result, giving the elements to be adjoined to a ring of the form k[x1, …,xn] to obtain its integral closure, is limited to the rather special case of the coordinate ring of a space curve all of whose singularities are normal. But in numerous examples where the curve has nonnormal singularities, the same method, which is essentially a modification of the method of locally quadratic transformations, also gives the integral closure.

Generalized k-Configurations

2005 ◽  
Vol 57 (2) ◽  
pp. 400-415
Author(s):  
Sindi Sabourin
Keyword(s):  
Projective Space ◽  
Complete Intersection ◽  
Hilbert Function ◽  
Complete Intersections ◽  
Dimensional Projective Space ◽  
Positive Integers

AbstractIn this paper, we find configurations of points in n-dimensional projective space (Pn) which simultaneously generalize both k-configurations and reduced 0-dimensional complete intersections. Recall that k-configurations in P2 are disjoint unions of distinct points on lines and in Pn are inductively disjoint unions of k-configurations on hyperplanes, subject to certain conditions. Furthermore, the Hilbert function of a k-configuration is determined from those of the smaller k-configurations. We call our generalized constructions kD-configurations, where D = {d1, … , dr} (a set of r positive integers with repetition allowed) is the type of a given complete intersection in Pn. We show that the Hilbert function of any kD-configuration can be obtained from those of smaller kD-configurations. We then provide applications of this result in two different directions, both of which are motivated by corresponding results about k-configurations.

scholarly journals Extensions of Toric Varieties

10.37236/580 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Mesut Şahin
Keyword(s):  
Local Ring ◽  
Complete Intersection ◽  
Toric Variety ◽  
Tangent Cone ◽  
Hilbert Function ◽  
Toric Varieties ◽  
Special Property ◽  
Fundamental Properties

In this paper, we introduce the notion of "extension" of a toric variety and study its fundamental properties. This gives rise to infinitely many toric varieties with a special property, such as being set theoretic complete intersection or arithmetically Cohen-Macaulay (Gorenstein) and having a Cohen-Macaulay tangent cone or a local ring with non-decreasing Hilbert function, from just one single example with the same property, verifying Rossi's conjecture for larger classes and extending some results appeared in literature.

scholarly journals Algebraic Properties of the Coordinate Ring of a Convex Polyomino

10.37236/7728 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Claudia Andrei
Keyword(s):  
Upper Bound ◽  
Recursive Formula ◽  
Coordinate Ring ◽  
Algebraic Properties ◽  
Coordinate Rings

 We classify all convex polyominoes whose coordinate rings are Gorenstein. We also give an upper bound for the Castelnuovo-Mumford regularity of the coordinate ring of any convex polyomino in terms of the smallest interval which contains its vertices. We give a recursive formula for computing the multiplicity of a stack polyomino.

scholarly journals The Gorenstein Property for Projective Coordinate Rings of the Moduli of Parabolic $\mathrm{SL}_2$-Principal Bundles on a Smooth Curve

10.37236/6438 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Theodore Faust ◽  
Christopher Manon
Keyword(s):  
Smooth Curve ◽  
Coordinate Ring ◽  
Projective Curve ◽  
Principal Bundles ◽  
Coordinate Rings ◽  
Marked Points ◽  
Combinatorial Methods ◽  
Projective Coordinate

Using combinatorial methods, we determine that a projective coordinate ring of the moduli of parabolic principal $\mathrm{SL}_2-$bundles on a marked projective curve is not Gorenstein when the genus and number of marked points are greater than $1$.

Fat points, partial intersections and Hamming distance

2019 ◽  
Vol 19 (04) ◽  
pp. 2050071 ◽  
Author(s):  
Susan M. Cooper ◽  
Elena Guardo
Keyword(s):  
Complete Intersection ◽  
Linear Code ◽  
Hilbert Function ◽  
Hamming Distance ◽  
Special Linear ◽  
Fat Points ◽  
Fat Point ◽  
Minimum Hamming Distance ◽  
Point Set ◽  
Set Of Points

We use two main techniques, namely, residuation and separators of points, to show that the Hilbert function of a certain fat point set supported on a grid complete intersection is the same as the Hilbert function of a reduced set of points called a partial intersection. As an application, we answer a question of Tohǎneanu and Van Tuyl which relates the minimum Hamming distance of a special linear code and the minimum socle degree of the associated fat point set.

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