Homogeneous Coordinate Ring Research Articles

Dimer models and Grassmannian cluster categories

AbstractA dimer model can be defined as a quiver embedded into a surface in such a way that the complement is a disjoint union of disks with oriented boundaries. Such models can also be considered in the case of a surface with boundary. The Postnikov diagrams used by J. Scott to describe the cluster structure of the homogeneous coordinate ring of the Grassmannian give rise to dimer models on a disk in this sense. We associate a natural algebra to such a dimer model. This algebra is a modified version of the corresponding Jacobian algebra, taking the boundary into account. Taking the sum of the idempotents corresponding to boundary vertices, we obtain an idempotent subalgebra, which we call the boundary algebra. We show that it is independent of the choice of dimer model and coincides with an algebra that B. Jensen, A. King and X. Su have used to model the cluster structure of the homogeneous coordinate ring of the Grassmannian categorically. This reports on joint work with A. King (Bath) and R. Marsh (Leeds) and with D. Bogdanic (Graz). (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

BLOWUPS AND FIBERS OF MORPHISMS

Our object of study is a rational map defined by homogeneous forms $g_{1},\ldots ,g_{n}$, of the same degree $d$, in the homogeneous coordinate ring $R=k[x_{1},\ldots ,x_{s}]$ of $\mathbb{P}_{k}^{s-1}$. Our goal is to relate properties of $\unicode[STIX]{x1D6F9}$, of the homogeneous coordinate ring $A=k[g_{1},\ldots ,g_{n}]$ of the variety parameterized by $\unicode[STIX]{x1D6F9}$, and of the Rees algebra ${\mathcal{R}}(I)$, the bihomogeneous coordinate ring of the graph of $\unicode[STIX]{x1D6F9}$. For a regular map $\unicode[STIX]{x1D6F9}$, for instance, we prove that ${\mathcal{R}}(I)$ satisfies Serre’s condition $R_{i}$, for some $i>0$, if and only if $A$ satisfies $R_{i-1}$ and $\unicode[STIX]{x1D6F9}$ is birational onto its image. Thus, in particular, $\unicode[STIX]{x1D6F9}$ is birational onto its image if and only if ${\mathcal{R}}(I)$ satisfies $R_{1}$. Either condition has implications for the shape of the core, namely, $\text{core}(I)$ is the multiplier ideal of $I^{s}$ and $\text{core}(I)=(x_{1},\ldots ,x_{s})^{sd-s+1}.$ Conversely, for $s=2$, either equality for the core implies birationality. In addition, by means of the generalized rows of the syzygy matrix of $g_{1},\ldots ,g_{n}$, we give an explicit method to reduce the nonbirational case to the birational one when $s=2$.

Characterizations of zero-dimensional complete intersections

Given a 0-dimensional subscheme \({\mathbb X}\) of a projective space \({\mathbb P}^n_K\) over a field K, we characterize in different ways whether \({\mathbb X}\) is the complete intersection of n hypersurfaces. Besides a generalization of the notion of a Cayley–Bacharach scheme, these characterizations involve the Kahler and the Dedekind different of the homogeneous coordinate ring of \({\mathbb X}\) or its Artinian reduction. We also characterize arithmetically Gorenstein schemes in novel ways and bring in further tools such as the module of regular differential forms, the fundamental class, and the Jacobian module of \({\mathbb X}\). Throughout we strive to work over an arbitrary base field K and keep the scheme \({\mathbb X}\) as general as possible, thereby improving several known characterizations.

A categorification of Grassmannian cluster algebras

We describe a ring whose category of Cohen-Macaulay modules provides an additive categorification of the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian of k-planes in n-space. More precisely, there is a cluster character defined on the category which maps the rigid indecomposable objects to the cluster variables and the maximal rigid objects to clusters. This is proved by showing that the quotient of this category by a single projective-injective object is Geiss-Leclerc-Schroer's category Sub $Q_k$, which categorifies the coordinate ring of the big cell in this Grassmannian.

Newton–Okounkov convex bodies of Schubert varieties and polyhedral realizations of crystal bases

A Newton-Okounkov convex body is a convex body constructed from a projective variety with a valuation on its homogeneous coordinate ring; this is deeply connected with representation theory. For instance, the Littelmann string polytopes and the Feigin-Fourier-Littelmann-Vinberg polytopes are examples of Newton-Okounkov convex bodies. In this paper, we prove that the Newton-Okounkov convex body of a Schubert variety with respect to a specific valuation is identical to the Nakashima-Zelevinsky polyhedral realization of a Demazure crystal. As an application of this result, we show that Kashiwara's involution (*-operation) corresponds to a change of valuations on the rational function field.

Multigraded Hilbert functions and toric complete intersection codes

Let X be a complete n-dimensional simplicial toric variety with homogeneous coordinate ring S. We study the multigraded Hilbert function HY of reduced 0-dimensional subschemes Y in X. We provide explicit formulas and prove non-decreasing and stabilization properties of HY when Y is a 0-dimensional complete intersection in X. We apply our results to computing the dimension of some evaluation codes on 0-dimensional complete intersection in simplicial toric varieties.

Cocycle twists of 4-dimensional Sklyanin algebras

We study cocycle twists of a 4-dimensional Sklyanin algebra A and a factor ring B which is a twisted homogeneous coordinate ring. Twisting such algebras by the Klein four-group G, we show that the twists AG,μ and BG,μ have very different geometric properties to their untwisted counterparts. For example, AG,μ has only 20 point modules and infinitely many fat point modules of multiplicity 2. The ring BG,μ falls under the purview of Artin and Stafford's classification of noncommutative curves, and we describe it using a sheaf of orders over an elliptic curve.

David Rees, FRS 1918–2013

David Rees, FRS 1918–2013

Twists of Plücker Coordinates as Dimer Partition Functions

The homogeneous coordinate ring of the Grassmannian Gr k,n has a cluster structure defined in terms of planar diagrams known as Postnikov diagrams. The cluster corresponding to such a diagram consists entirely of Plucker coordinates. We introduce a twist map on Gr k,n , related to the Berenstein–Fomin–Zelevinsky-twist, and give an explicit Laurent expansion for the twist of an arbitrary Plucker coordinate in terms of the cluster variables associated with a fixed Postnikov diagram. The expansion arises as a (scaled) dimer partition function of a weighted version of the bipartite graph dual to the Postnikov diagram, modified by a boundary condition determined by the Plucker coordinate. We also relate the twist map to a maximal green sequence.

Tate curves and UHF-algebras

It is proved that (a stabilization of) the norm-closure of a self-adjoint representation of the twisted homogeneous coordinate ring of a Tate curve contains a copy of the UHF-algebra.

Crystal bases and Newton–Okounkov bodies

Let G be a connected reductive algebraic group. We prove that the string parametrization of a crystal basis for a finite dimensional irreducible representation of G extends to a natural valuation on the field of rational functions on the flag variety G/B, which is a highest term valuation corresponding to a coordinate system on a Bott-Samelson variety. This shows that the string polytopes associated to irreducible representations, can be realized as Newton-Okounkov bodies for the flag variety. This is closely related to an earlier result of A. Okounkov for the Gelfand-Cetlin polytopes of the symplectic group. As a corollary we recover a multiplicativity property of the canonical basis due to P. Caldero. We generalize the results to spherical varieties. From these the existence of SAGBI bases for the homogeneous coordinate rings of flag and spherical varieties, as well as their toric degenerations follow, recovering previous results of Caldero, Alexeev-Brion and the author.

Dimers, webs, and positroids

We study the dimer model for a planar bipartite graph N embedded in a disk, with boundary vertices on the boundary of the disk. Counting dimer configurations with specified boundary conditions gives a point in the totally non-negative Grassmannian. Considering pairing probabilities for the double-dimer model gives rise to Grassmann analogs of Rhoades and Skandera's Temperley–Lieb immanants. The same problem for the (probably novel) triple-dimer model gives rise to the combinatorics of Kuperberg's webs and Grassmann analogs of Pylyavskyy's web immanants. This draws a connection between the square move of plabic graphs (or urban renewal of planar bipartite graphs), and Kuperberg's square reduction of webs. Our results also suggest that canonical-like bases might be applied to the dimer model. We furthermore show that these functions on the Grassmannian are compatible with restriction to positroid varieties. Namely, our construction gives bases for the degree 2 and degree 3 components of the homogeneous coordinate ring of a positroid variety that are compatible with the cyclic group action.

Standard Monomial Theory of RR Varieties

We construct the RR varieties as the fiber products of Bott-Samelson varieties over Richardson varieties. We study their homogeneous coordinate rings and standard monomial theory.

Quantum deformations of projective three-space

We describe the possible noncommutative deformations of complex projective three-space by exhibiting the Calabi–Yau algebras that serve as their homogeneous coordinate rings. We prove that the space parametrizing such deformations has exactly six irreducible components, and we give explicit presentations for the generic members of each family in terms of generators and relations. The proof uses deformation quantization to reduce the problem to a similar classification of unimodular quadratic Poisson structures in four dimensions, which we extract from Cerveau and Lins Neto's classification of degree-two foliations on projective space. Corresponding to the “exceptional” component in their classification is a quantization of the third symmetric power of the projective line that supports bimodule quantizations of the classical Schwarzenberger bundles.

Naïve blowups and canonical birationally commutative factors

In 2008, Rogalski and Zhang (Math Z 259(2):433–455, 2008) showed that if R is a strongly noetherian connected graded algebra over an algebraically closed field \({\Bbbk }\), then R has a canonical birationally commutative factor. This factor is, up to finite dimension, a twisted homogeneous coordinate ring \(B(X, \mathcal {L}, \sigma )\); here X is the projective parameter scheme for point modules over R, as well as tails of points in \({{\mathrm{Qgr-\!}}}R\). (As usual, \(\sigma \) is an automorphism of X, and \(\mathcal {L}\) is a \(\sigma \)-ample invertible sheaf on X). We extend this result to a large class of noetherian (but not strongly noetherian) algebras. Specifically, let R be a noetherian connected graded \({\Bbbk }\)-algebra, where \({\Bbbk }\) is an uncountable algebraically closed field. Let \(Y_\infty \) denote the parameter space (or stack or proscheme) parameterizing R-point modules, and suppose there is a projective variety X that corepresents tails of points. There is a canonical map p:\(Y_\infty \rightarrow X\). If the indeterminacy locus of \(p^{-1}\) is 0-dimensional and X satisfies a mild technical assumption, we show that there is a homomorphism g:\(R \rightarrow B(X,\mathcal {L},\sigma )\), and that g(R) is, up to finite dimension, a naive blowup on X in the sense of Keeler et al. (Duke Math J 126(3):491–546, 2005), Rogalski and Stafford (J Algebra 318(2):794–833, 2007) and satisfies a universal property. We further show that the point space \(Y_\infty \) is noetherian.

Kähler differentials and Kähler differents for fat point schemes

Given a fat point scheme W=m1P1+⋯+msPs in a projective space Pn over a field K, we study the module of Kähler differentials and the Kähler differents of its homogeneous coordinate ring RW. We describe the Hilbert functions and Hilbert polynomials of these objects and bound their index of regularity. For special cases, in particular if the support of W is a complete intersection or has some kind of uniformity, or if n=4, we present more detailed results, including proofs of the Segre bound for certain fat point schemes in P4.

On higher syzygies of ruled surfaces III

In this paper, we study the minimal free resolution of homogeneous coordinate rings of a ruled surface S over a curve of genus g with the numerical invariant e<0 and a minimal section C0. Let L∈PicX be a line bundle in the numerical class of aC0+bf such that a≥1 and 2b−ae=4g−1+k for some k≥max(2,−e). We prove that the Green–Lazarsfeld index index(S,L) of (S,L), i.e. the maximum p such that L satisfies condition N2,p, satisfies the inequalitiesk2−g≤index(S,L)≤k2−ae+32+max(0,⌈2g−3+ae−k4⌉). Also if S has an effective divisor D≡2C0+ef, then we obtain another upper bound of index(S,L), i.e., index(S,L)≤k+max(0,⌈2g−4−k2⌉). This gives a better bound in case b is small compared to a. Finally, for each e∈{−g,…,−1} we construct a ruled surface S with the numerical invariant e and a minimal section C0 which has an effective divisor D≡2C0+ef.

Integrable systems, toric degenerations and Okounkov bodies

Let $$X$$ be a smooth projective variety of dimension $$n$$ over $$\mathbb {C}$$ equipped with a very ample Hermitian line bundle $$\mathcal {L}$$ . In the first part of the paper, we show that if there exists a toric degeneration of $$X$$ satisfying some natural hypotheses (which are satisfied in many settings), then there exists a surjective continuous map from $$X$$ to the special fiber $$X_0$$ which is a symplectomorphism on an open dense subset $$U$$ . From this we are then able to construct a completely integrable system on $$X$$ in the sense of symplectic geometry. More precisely, we construct a collection of real-valued functions $$\{H_1, \ldots , H_n\}$$ on $$X$$ which are continuous on all of $$X$$ , smooth on an open dense subset $$U$$ of $$X$$ , and pairwise Poisson-commute on $$U$$ . Moreover, our integrable system in fact generates a Hamiltonian torus action on $$U$$ . In the second part, we show that the toric degenerations arising in the theory of Newton-Okounkov bodies satisfy all the hypotheses of the first part of the paper. In this case the image of the ‘moment map’ $$\mu = (H_1, \ldots , H_n): X \rightarrow \mathbb {R}^n$$ is precisely the Newton-Okounkov body $$\Delta = \Delta (R, v)$$ associated to the homogeneous coordinate ring $$R$$ of $$X$$ , and an appropriate choice of a valuation $$v$$ on $$R$$ . Our main technical tools come from algebraic geometry, differential (Kähler) geometry, and analysis. Specifically, we use the gradient-Hamiltonian vector field, and a subtle generalization of the famous Łojasiewicz gradient inequality for real-valued analytic functions. Since our construction is valid for a large class of projective varieties $$X$$ , this manuscript provides a rich source of new examples of integrable systems. We discuss concrete examples, including elliptic curves, flag varieties of arbitrary connected complex reductive groups, spherical varieties, and weight varieties.

Cohen-Macaulayness of Rees algebras of diagonal ideals

Given two determinantal rings over a field $k$, we consider the Rees algebra of the diagonal ideal, the kernel of the multiplication map. The special fiber ring of the diagonal ideal is the homogeneous coordinate ring of the secant variety. When the Rees algebra and the symmetric algebra coincide, we show that the Rees algebra is Cohen-Macaulay.

Filtrations and test-configurations

We introduce a strengthening of K-stability, based on filtrations of the homogeneous coordinate ring. This allows for considering certain limits of families of test-configurations, which arise naturally in several settings. We prove that if a manifold with no automorphisms admits a cscK metric, then it satisfies this stronger stability notion. We also discuss the relation with the birational transformations in the definition of \(b\)-stability.