Homogeneous Coordinate Ring Research Articles

Hilbert–Kunz theory for nodal cubics, via sheaves

Suppose B = F [ x , y , z ] / h is the homogeneous coordinate ring of a characteristic p degree 3 irreducible plane curve C with a node. Let J be a homogeneous ( x , y , z ) -primary ideal and n → e n be the Hilbert–Kunz function of B with respect to J. Let q = p n . When J = ( x , y , z ) , it is known that e n = 7 3 q 2 − 1 3 q − R where R = 5 3 if q ≡ 2 ( 3 ) , and is 1 otherwise. We generalize this, showing that e n = μ q 2 + α q − R where R only depends on q mod 3 . We describe α and R in terms of classification data for a vector bundle on C.

Noncommutative complex geometry of the quantum projective space

We define holomorphic structures on canonical line bundles of the quantum projective space C P q ℓ and identify their space of holomorphic sections. This determines the quantum homogeneous coordinate ring of the quantum projective space. We show that the fundamental class of C P q ℓ is naturally presented by a twisted positive Hochschild cocycle. Finally, we verify the main statements of the Riemann–Roch formula and the Serre duality for C P q 1 and C P q 2 .

Rees algebras of diagonal ideals

There is a natural epimorphism from the symmetric algebra to the Rees algebra of an ideal. When this epimorphism is an isomorphism, we say that the ideal is of linear type. Given two determinantal rings over a field, we consider the diagonal ideal, kernel of the multiplication map. We prove in many cases that the diagonal ideal is of linear type and recover the defining ideal of the Rees algebra. In our cases, the special fiber rings of the diagonal ideals are the homogeneous coordinate rings of the join varieties.

Rational normal scrolls and the defining equations of Rees algebras

Consider a height two ideal, I, which is minimally generated by m homogeneous forms of degree d in the polynomial ring R = k[x, y]. Suppose that one column in the homogeneous presenting matrix φ of I has entries of degree n and all of the other entries of φ are linear. We identify an explicit generating set for the ideal which defines the Rees algebra ℛ = R[It]; so for the polynomial ring S = R[T1, . . . , Tm]. We resolve ℛ as an S-module and Is as an R-module, for all powers s. The proof uses the homogeneous coordinate ring, A = S/H, of a rational normal scroll, with . The ideal is isomorphic to the nth symbolic power of a height one prime ideal K of A. The ideal K(n) is generated by monomials. Whenever possible, we study A/K(n) in place of because the generators of K(n) are much less complicated then the generators of . We obtain a filtration of K(n) in which the factors are polynomial rings, hypersurface rings, or modules resolved by generalized Eagon–Northcott complexes. The generators of I parameterize an algebraic curve in projective m – 1 space. The defining equations of the special fiber ring ℛ/(x, y)ℛ yield a solution of the implicitization problem for .

The Dixmier-Moeglin equivalence for twisted homogeneous coordinate rings

Given a projective scheme X over a field k, an automorphism σ: X → X, and a σ-ample invertible sheaf L, one may form the twisted homogeneous coordinate ring B = B(X, L, σ), one of the most fundamental constructions in noncommutative projective algebraic geometry. We study the primitive spectrum of B, as well as that of other closely related algebras such as skew and skew-Laurent extensions of commutative algebras. Over an algebraically closed, uncountable field k of characteristic zero, we prove that the primitive ideals of B are characterized by the usual Dixmier-Moeglin conditions whenever dim X ≤ 2.

The homogeneous coordinate ring of the quantum projective plane

We define holomorphic structures on canonical line bundles on the quantum projective plane. The space of holomorphic sections of these line bundles will determine the quantum homogeneous coordinate ring of C P q 2 . We also show that the holomorphic structure of C P q 2 is naturally represented by a twisted positive Hochschild 4-cocycle.

COMPUTATION OF THE GROTHENDIECK AND PICARD GROUPS OF A TORIC DM STACK BY USING A HOMOGENEOUS COORDINATE RING FOR

AbstractWe compute the Grothendieck and Picard groups of a smooth toric DM stack by using a suitable category of graded modules over a polynomial ring. The polynomial ring with a suitable grading and suitable irrelevant ideal functions is a homogeneous coordinate ring for the stack.

Geometric idealizer rings

Let X be a projective variety, $\sigma$ an automorphism of X, L a $\sigma$-ample invertible sheaf on X, and Z a closed subscheme of X. Inside the twisted homogeneous coordinate ring $B = B(X, L, \sigma)$, let I be the right ideal of sections vanishing at Z. We study the subring R = k + I of B. Under mild conditions on Z and $\sigma$, R is the idealizer of I in B: the maximal subring of B in which I is a two-sided ideal. We give geometric conditions on Z and $\sigma$ that determine the algebraic properties of R, and show that if Z and $\sigma$ are sufficiently general, in a sense we make precise, then R is left and right noetherian, has finite left and right cohomological dimension, is strongly right noetherian but not strongly left noetherian, and satisfies right $\chi_d$ (where d = \codim Z) but fails left $\chi_1$. We also give an example of a right noetherian ring with infinite right cohomological dimension, partially answering a question of Stafford and Van den Bergh. This generalizes results of Rogalski in the case that Z is a point in $\mathbb{P}^d$.

Annihilators of graded components of the canonical module, and the core of standard graded algebras

We relate the annihilators of graded components of the canonical module of a graded Cohen-Macaulay ring to colon ideals of powers of the homogeneous maximal ideal. In particular, we connect them to the core of the maximal ideal. An application of our results characterizes Cayley-Bacharach sets of points in terms of the structure of the core of the maximal ideal of their homogeneous coordinate ring. In particular, we show that a scheme is Cayley-Bacharach if and only if the core is a power of the maximal ideal.

Geometric algebras on projective surfaces

Let X be a projective surface, let \sigma be an automorphism of X, and let L be a \sigma-ample invertible sheaf on X. We study the properties of a family of subrings, parameterized by geometric data, of the twisted homogeneous coordinate ring B(X, L, \sigma). In particular, we find necessary and sufficient conditions for these subrings to be noetherian. We also study their homological properties, their associated noncommutative projective schemes, and when they are maximal orders. In the process, we produce new examples of maximal orders; these are graded and have the property that no Veronese subring is generated in degree 1. Our results are used in a companion paper to give defining data for a large class of noncommutative projective surfaces.

Holomorphic Structures on the Quantum Projective Line

We show that much of the structure of the 2-sphere as a complex curve survives the q-deformation and has natural generalizations to the quantum 2-sphere - which, with additional structures, we identify with the quantum projective line. Notably among these is the identification of a quantum homogeneous coordinate ring with the coordinate ring of the quantum plane. In parallel with the fact that positive Hochschild cocycles on the algebra of smooth functions on a compact oriented 2-dimensional manifold encode the information for complex structures on the surface, we formulate a notion of twisted positivity for twisted Hochschild and cyclic cocycles and exhibit an explicit twisted positive Hochschild cocycle for the complex structure on the sphere.

Equivariant primary decomposition and toric sheaves

We study global primary decompositions in the category of sheaves on a scheme which are equivariant under the action of an algebraic group. We show that equivariant primary decompositions exist if the group is connected. As main application we consider the case of varieties which are quotients of a quasi-affine variety by the action of a diagonalizable group and thus admit a homogeneous coordinate ring, such as toric varieties. Comparing these decompositions with primary decompositions of graded modules over the homogeneous coordinate ring, we show that these are equivalent if the action of the diagonalizable group is free. We give some specific examples for the case of toric varieties.

EQUATIONS FOR $\overline M_{0,n}$

We show that the log canonical bundle, κ, of [Formula: see text] is very ample, show the homogeneous coordinate ring is Koszul, and give a nice set of rank 4 quadratic generators for the homogeneous ideal: The embedding is equivariant for the symmetric group, and the image lies on many Segre embedded copies of ℙ1 × ⋯ × ℙn-3, permuted by the symmetric group. The homogeneous ideal of [Formula: see text] is the sum of the homogeneous ideals of these Segre embeddings.

Divisors on rational normal scrolls

Let A be the homogeneous coordinate ring of a rational normal scroll. The ring A is equal to the quotient of a polynomial ring S by the ideal generated by the two by two minors of a scroll matrix ψ with two rows and ℓ catalecticant blocks. The class group of A is cyclic, and is infinite provided ℓ is at least two. One generator of the class group is [ J ] , where J is the ideal of A generated by the entries of the first column of ψ. The positive powers of J are well-understood; if ℓ is at least two, then the nth ordinary power, the nth symmetric power, and the nth symbolic power coincide and therefore all three nth powers are resolved by a generalized Eagon–Northcott complex. The inverse of [ J ] in the class group of A is [ K ] , where K is the ideal generated by the entries of the first row of ψ. We study the positive powers of [ K ] . We obtain a minimal generating set and a Gröbner basis for the preimage in S of the symbolic power K ( n ) . We describe a filtration of K ( n ) in which all of the factors are Cohen–Macaulay S-modules resolved by generalized Eagon–Northcott complexes. We use this filtration to describe the modules in a finely graded resolution of K ( n ) by free S-modules. We calculate the regularity of the graded S-module K ( n ) and we show that the symbolic Rees ring of K is Noetherian.

Degenerate Sklyanin algebras and generalized twisted homogeneous coordinate rings

In this work, we introduce the point parameter ring B, a generalized twisted homogeneous coordinate ring associated to a degenerate version of the three-dimensional Sklyanin algebra. The surprising geometry of these algebras yields an analogue to a result of Artin–Tate–van den Bergh, namely that B is generated in degree one and thus is a factor of the corresponding degenerate Sklyanin algebra.

Arithmetic Structures on Noncommutative Tori with Real Multiplication

We study the homogeneous coordinate rings of real multiplication noncommutative tori as defined in (23). Our aim is to understand how these rings give rise to an arithmetic structure on the noncommutative torus. We start by giving an explicit presentation of these rings in terms of their natural generators. This presentation gives us information about the rationality of the rings and is the starting point for the analysis of the corresponding geometric data. Linear bases for these rings are also discussed. The basic role played by theta functions in these computations gives relations with elliptic curves at the level of modular curves. We use the modularity of these coordinate rings to obtain algebras which do no depend on the choice of a complex structure on the noncommutative torus. These algebras are obtained by an averaging process over (limiting) modular symbols.

The AS–Cohen–Macaulay property for quantum flag manifolds of minuscule weight

It is shown that quantum homogeneous coordinate rings of generalised flag manifolds corresponding to minuscule weights, their Schubert varieties, big cells, and determinantal varieties are AS–Cohen–Macaulay. The main ingredient in the proof is the notion of a quantum graded algebra with a straightening law, introduced by T.H. Lenagan and L. Rigal [T.H. Lenagan, L. Rigal, Quantum graded algebras with a straightening law and the AS–Cohen–Macaulay property for quantum determinantal rings and quantum Grassmannians, J. Algebra 301 (2006) 670–702]. Using Stanley's Theorem it is moreover shown that quantum generalised flag manifolds of minuscule weight and their big cells are AS–Gorenstein.

On tangential varieties of rational homogeneous varieties

The sphericality of tangential varieties of rational homogeneous varieties is determined. The homogeneous coordinate rings and rings of covariants of the tangential varieties of homogenously embedded compact Hermitian symmetric spaces (CHSS) are determined. Bounds on the degrees of generators of the ideals of tangential varieties of CHSS are given, and more explicit information is obtained about the ideals in certain cases.

A Sufficient Condition for a Hibi Ring to Be Level and Levelness of Schubert Cycles

Let K be a field, D a finite distributive lattice and P the set of all join-irreducible elements of D. We show that if {y ∈ P | y ≥ x} is pure for any x ∈ P, then the Hibi ring ℛ K (D) is level. Using this result and the argument of sagbi basis theory, we show that the homogeneous coordinate rings of Schubert subvarieties of Grassmannians are level.

Universal torsors of Del Pezzo surfaces and homogeneous spaces

Let Cox ( S r ) be the homogeneous coordinate ring of the blow-up S r of P 2 in r general points, i.e., a smooth Del Pezzo surface of degree 9 − r . We prove that for r ∈ { 6 , 7 } , Proj ( Cox ( S r ) ) can be embedded into G r / P r , where G r is an algebraic group with root system given by the primitive Picard lattice of S r and P r ⊂ G r is a certain maximal parabolic subgroup.