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 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.

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 The solvable length of groups of local diffeomorphisms

2019 ◽  
Vol 2019 (752) ◽  
pp. 105-139 ◽  
Author(s):  
Javier Ribón
Keyword(s):  
Upper Bound ◽  
Solvable Groups ◽  
Ambient Space ◽  
Natural Question ◽  
Local Diffeomorphisms ◽  
Derived Length ◽  
Local Complex ◽  
Algebraic Properties ◽  
Solvable Length

Abstract We are interested in the algebraic properties of groups of local biholomorphisms and their consequences. A natural question is whether the complexity of solvable groups is bounded by the dimension of the ambient space. In this spirit we show that {2n+1} is the sharpest upper bound for the derived length of solvable subgroups of the group {\mathrm{Diff}({\mathbb{C}}^{n},0)} of local complex analytic diffeomorphisms for {n=2,3,4,5} .

Algebraic Properties of Modulo q Complete ℓ-Wide Families

2009 ◽  
Vol 18 (3) ◽  
pp. 309-333 ◽  
Author(s):  
BÁLINT FELSZEGHY ◽  
GÁBOR HEGEDŰS ◽  
LAJOS RÓNYAI
Keyword(s):  
Set Theory ◽  
Upper Bound ◽  
Gröbner Basis ◽  
Hilbert Function ◽  
Grobner Basis ◽  
Vanishing Ideal ◽  
Algebraic Properties ◽  
Image Position ◽  
Extremal Set ◽  
Wide Family

Let q be a power of a prime p, and let n, d, ℓ be integers such that 1 ≤ n, 1 ≤ ℓ < q. Consider the modulo q complete ℓ-wide family: We describe a Gröbner basis of the vanishing ideal I() of the set of characteristic vectors of over fields of characteristic p. It turns out that this set of polynomials is a Gröbner basis for all term orderings ≺, for which the order of the variables is xn ≺ xn−1 ≺ ⋅⋅⋅ ≺ x1.We compute the Hilbert function of I(), which yields formulae for the modulo p rank of certain inclusion matrices related to .We apply our results to problems from extremal set theory. We prove a sharp upper bound of the cardinality of a modulo q ℓ-wide family, which shatters only small sets. This is closely related to a conjecture of Frankl [13] on certain ℓ-antichains. The formula of the Hilbert function also allows us to obtain an upper bound on the size of a set system with certain restricted intersections, generalizing a bound proposed by Babai and Frankl [6].The paper generalizes and extends the results of [15], [16] and [17].

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 Homological invariants of the Stanley–Reisner ring of a k-decomposable simplicial complex

2017 ◽  
Vol 10 (03) ◽  
pp. 1750061
Author(s):  
Somayeh Moradi
Keyword(s):  
Simplicial Complex ◽  
Upper Bound ◽  
Betti Numbers ◽  
Recursive Formula ◽  
Projective Dimension ◽  
Simplicial Complexes ◽  
Monomial Ideals ◽  
Edge Ideals ◽  
Graded Betti Numbers ◽  
Shellable Simplicial Complex

In this paper, we study the regularity and the projective dimension of the Stanley–Reisner ring of a [Formula: see text]-decomposable simplicial complex and explain these invariants with a recursive formula. To this aim, the graded Betti numbers of decomposable monomial ideals which is the dual concept for [Formula: see text]-decomposable simplicial complexes are studied and an inductive formula for the Betti numbers is given. As a corollary, for a shellable simplicial complex [Formula: see text], a formula for the regularity of the Stanley–Reisner ring of [Formula: see text] is presented. Finally, for a chordal clutter [Formula: see text], an upper bound for [Formula: see text] is given in terms of the regularities of edge ideals of some chordal clutters which are minors of [Formula: see text].

scholarly journals The cd-index of Bruhat Intervals

10.37236/1827 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Nathan Reading
Keyword(s):  
Upper Bound ◽  
Recursive Formula ◽  
Bruhat Order ◽  
Boolean Algebras ◽  
Structural Recursion ◽  
Simple Effect ◽  
The Face ◽  
Stacked Polytope ◽  
Special Case ◽  
Bruhat Intervals

We study flag enumeration in intervals in the Bruhat order on a Coxeter group by means of a structural recursion on intervals in the Bruhat order. The recursion gives the isomorphism type of a Bruhat interval in terms of smaller intervals, using basic geometric operations which preserve PL sphericity and have a simple effect on the cd-index. This leads to a new proof that Bruhat intervals are PL spheres as well a recursive formula for the cd-index of a Bruhat interval. This recursive formula is used to prove that the cd-indices of Bruhat intervals span the space of cd-polynomials. The structural recursion leads to a conjecture that Bruhat spheres are "smaller" than polytopes. More precisely, we conjecture that if one fixes the lengths of $x$ and $y$, then the cd-index of a certain dual stacked polytope is a coefficientwise upper bound on the cd-indices of Bruhat intervals $[x,y]$. We show that this upper bound would be tight by constructing Bruhat intervals which are the face lattices of these dual stacked polytopes. As a weakening of a special case of the conjecture, we show that the flag h-vectors of lower Bruhat intervals are bounded above by the flag h-vectors of Boolean algebras (i. e. simplices).

scholarly journals Inclusion ideals associated to uniformly increasing hypergraphs

2013 ◽  
Vol 50 (2) ◽  
pp. 199-206
Author(s):  
I. Anwar ◽  
S. Ahmad ◽  
A. Inam ◽  
A. Haider
Keyword(s):  
Upper Bound ◽  
Special Class ◽  
Uniform Hypergraphs ◽  
Algebraic Properties

In this paper, we introduce inclusion ideals I(H) associated to a special class of non uniform hypergraphs H(gC; ɛ; d), namely, the uniformly increasing hypergraphs. We discuss some algebraic properties of the inclusion ideals. In particular, we give an upper bound of the Castelnouvo-Mumford regularity of the special dual ideal I[*](H).

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