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 $1$.

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.

A Torelli theorem for moduli spaces of principal bundles on curves defined over ℝ

2017 ◽  
Vol 28 (06) ◽  
pp. 1750049
Author(s):  
Indranil Biswas ◽  
Olivier Serman
Keyword(s):  
Algebraic Group ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Isomorphism Class ◽  
Projective Curve ◽  
Real Numbers ◽  
Principal Bundles ◽  
Smooth Projective Curve ◽  
Simple Factor ◽  
Torelli Theorem

Let [Formula: see text] be a geometrically irreducible smooth projective curve, of genus at least three, defined over the field of real numbers. Let [Formula: see text] be a connected reductive affine algebraic group, defined over [Formula: see text], such that [Formula: see text] is nonabelian and has one simple factor. We prove that the isomorphism class of the moduli space of principal [Formula: see text]-bundles on [Formula: see text] determine uniquely the isomorphism class of [Formula: see text].

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 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.

Stability of bundles with a given filtration on a smooth curve

1999 ◽  
Vol 129 (2) ◽  
pp. 229-234
Author(s):  
E. Ballico
Keyword(s):  
Vector Bundles ◽  
Smooth Curve ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Stable Vector Bundles

LetXbe a smooth projective curve of genusg ≥2. Here we construct stable vector bundles onXequipped with a filtration with suitable numerical properties and with stable graded subquotients.

Principal bundles over a smooth real projective curve of genus zero

2008 ◽  
Vol 8 (3) ◽  
Author(s):  
Indranil Biswas ◽  
Johannes Huisman
Keyword(s):  
Projective Curve ◽  
Genus Zero ◽  
Principal Bundles

The Convex Polytopes and Homogeneous Coordinate Rings of Bivariate Polynomials

2019 ◽  
Vol 16 (2) ◽  
pp. 1
Author(s):  
Shamsatun Nahar Ahmad ◽  
Nor’Aini Aris ◽  
Azlina Jumadi
Keyword(s):  
Convex Polytopes ◽  
Polynomial Systems ◽  
Matrix Formulation ◽  
Bivariate Polynomials ◽  
Homogeneous Coordinates ◽  
The Matrix ◽  
Projective Vector ◽  
Coordinate Rings ◽  
Resultant Matrix ◽  
The Given

Concepts from algebraic geometry such as cones and fans are related to toric varieties and can be applied to determine the convex polytopes and homogeneous coordinate rings of multivariate polynomial systems. The homogeneous coordinates of a system in its projective vector space can be associated with the entries of the resultant matrix of the system under consideration. This paper presents some conditions for the homogeneous coordinates of a certain system of bivariate polynomials through the construction and implementation of the Sylvester-Bèzout hybrid resultant matrix formulation. This basis of the implementation of the Bèzout block applies a combinatorial approach on a set of linear inequalities, named 5-rule. The inequalities involved the set of exponent vectors of the monomials of the system and the entries of the matrix are determined from the coefficients of facets variable known as brackets. The approach can determine the homogeneous coordinates of the given system and the entries of the Bèzout block. Conditions for determining the homogeneous coordinates are also given and proven.

Smooth curve drawing using a Bessel function

2015 ◽  
Author(s):  
Y. Q. Du ◽  
M. J. Pan ◽  
Q. Li ◽  
L. Li
Keyword(s):  
Bessel Function ◽  
Smooth Curve
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