Homogeneous Coordinate Ring Research Articles

Polynomial log-volume growth and the GK-dimensions of twisted homogeneous coordinate rings

Let f be a zero entropy automorphism of a compact Kähler manifold X . We study the polynomial log-volume growth \mathrm{Plov}(f) of f in light of the dynamical filtrations introduced in our previous work with T.-C. Dinh. We obtain new upper bounds and lower bounds of \mathrm{Plov}(f) . As a corollary, we completely determine \mathrm{Plov}(f) when \dim X = 3 , extending a result of Artin–Van den Bergh for surfaces. When X is projective, \mathrm{Plov}(f) + 1 coincides with the Gelfand–Kirillov dimensions \mathrm{GKdim}(X,f) of the twisted homogeneous coordinate rings associated to (X,f) . Reformulating these results for \mathrm{GKdim}(X,f) , we improve Keeler’s bounds of \mathrm{GKdim}(X,f) and provide effective upper bounds of \mathrm{GKdim}(X,f) which only depend on \dim X .

Differential theory of zero-dimensional schemes

To study a 0-dimensional scheme X in Pn over a perfect field K, we use the module of Kähler differentials ΩR/K1 of its homogeneous coordinate ring R and its exterior powers, the higher modules of Kähler differentials ΩR/Km. One of our main results is a characterization of weakly curvilinear schemes X by the Hilbert polynomials of the modules ΩR/Km which allows us to check this property algorithmically without computing the primary decomposition of the vanishing ideal of X. Further main achievements are precise formulas for the Hilbert functions and Hilbert polynomials of the modules ΩR/Km for a fat point scheme X which extend and settle previous partial results and conjectures. Underlying these results is a novel method: we first embed the homogeneous coordinate ring R into its truncated integral closure R˜. Then we use the corresponding map from the module of Kähler differentials ΩR/K1 to ΩR˜/K1 to find a formula for the Hilbert polynomial HP(ΩR/K1) and a sharp bound for the regularity index ri(ΩR/K1). Next we extend this to formulas for the Hilbert polynomials HP(ΩR/Km) and bounds for the regularity indices of the higher modules of Kähler differentials. As a further application, we characterize uniformity conditions on X using the Hilbert functions of the Kähler differential modules of X and its subschemes.

Polynomial Volume Growth of Quasi-Unipotent Automorphisms of Abelian Varieties (with an Appendix in Collaboration with Chen Jiang)

Abstract Let $X$ be an abelian variety over an algebraically closed field $\textbf{k}$ and $f$ a quasi-unipotent automorphism of $X$. When $\textbf{k}$ is the field of complex numbers, Lin, Oguiso, and D.-Q. Zhang provide an explicit formula for the polynomial volume growth of (or equivalently, for the Gelfand–Kirillov dimension of the twisted homogeneous coordinate ring associated with) the pair $(X, f)$, by an analytic argument. We give an algebraic proof of this formula that works in arbitrary characteristic. In the course of the proof, we obtain the following: (1) a new description of the action of endomorphisms on the $\ell $-adic Tate spaces, in comparison with recent results of Zarhin and Poonen–Rybakov; (2) a partial converse to a result of Reichstein, Rogalski, and J.J. Zhang on quasi-unipotency of endomorphisms and their pullback action on the rational Néron–Severi space $\textsf{N}^{1}(X)_{\textbf{Q}}$ of $\textbf{Q}$-divisors modulo numerical equivalence; and (3) the maximum size of Jordan blocks of (the Jordan canonical form of) $f^{*}|_{\textsf{N}^{1}(X)_{\textbf{Q}}}$ in terms of the action of $f$ on the Tate space $V_{\ell }(X)$.

Monomial projections of Veronese varieties: New results and conjectures

In this paper, we consider the homogeneous coordinate rings A(Yn,d)≅K[Ωn,d] of monomial projections Yn,d of Veronese varieties parameterized by subsets Ωn,d of monomials of degree d in n+1 variables where: (1) Ωn,d contains all monomials supported in at most s variables and, (2) Ωn,d is a set of monomial invariants of a finite diagonal abelian group G⊂GL(n+1,K) of order d. Our goal is to study when K[Ωn,d] is a quadratic algebra and, if so, when K[Ωn,d] is Koszul or G-quadratic. For the family (1), we prove that K[Ωn,d] is quadratic when s≥⌈n+22⌉. For the family (2), we completely characterize when K[Ω2,d] is quadratic in terms of the group G⊂GL(3,K), and we prove that K[Ω2,d] is quadratic if and only if it is Koszul. We also provide large families of examples where K[Ωn,d] is G-quadratic.

Classification and geometric properties of surfaces with property N3,3

Let X be a closed subscheme of codimension e in a projective space. One says that X satisfies property Nd,p, if the i-th syzygies of the homogeneous coordinate ring are generated by elements of degree <d+i for 0≤i≤p. The geometric and algebraic properties of smooth projective varieties satisfying property N2,e are well understood, and the complete classification of these varieties is a classical result. The aim of this paper is to study the next case: projective surfaces in P5 satisfying property N3,3. In particular, we give a classification of such varieties using adjunction mappings and we also provide illuminating examples of our results via calculations done with Macaulay 2. As corollaries, we study the CI-biliaison equivalence class of smooth projective surfaces of degree 10 satisfying property N3,3 on a cubic fourfold.

GENERIC LINES IN PROJECTIVE SPACE AND THE KOSZUL PROPERTY

AbstractIn this paper, we study the Koszul property of the homogeneous coordinate ring of a generic collection of lines in $\mathbb {P}^n$ and the homogeneous coordinate ring of a collection of lines in general linear position in $\mathbb {P}^n.$ We show that if $\mathcal {M}$ is a collection of m lines in general linear position in $\mathbb {P}^n$ with $2m \leq n+1$ and R is the coordinate ring of $\mathcal {M},$ then R is Koszul. Furthermore, if $\mathcal {M}$ is a generic collection of m lines in $\mathbb {P}^n$ and R is the coordinate ring of $\mathcal {M}$ with m even and $m +1\leq n$ or m is odd and $m +2\leq n,$ then R is Koszul. Lastly, we show that if $\mathcal {M}$ is a generic collection of m lines such that $$ \begin{align*} m&gt; \frac{1}{72}\left(3(n^2+10n+13)+\sqrt{3(n-1)^3(3n+5)}\right),\end{align*} $$ then R is not Koszul. We give a complete characterization of the Koszul property of the coordinate ring of a generic collection of lines for $n \leq 6$ or $m \leq 6$ . We also determine the Castelnuovo–Mumford regularity of the coordinate ring for a generic collection of lines and the projective dimension of the coordinate ring of collection of lines in general linear position.

Mukai Conjecture

In algebraic geometry, syzygies roughly refers to the relations of defining equations of a projective variety embedded in a projective space. They are defined in terms of a minimal graded free resolution for the homogeneous coordinate ring over a polynomial ring. Since the resolution is unique up to isomorphism, it makes sense to talk about the number of a minimal set of generators of given degree in any syzygy module.

The Frobenius morphism in invariant theory II

Let R be the homogeneous coordinate ring of the Grassmannian G=Gr(2,n) defined over an algebraically closed field k of characteristic p≥max⁡{n−2,3}. In this paper we give a description of the decomposition of R, considered as graded Rpr-module, for r≥2. This is a companion paper to [16], where the case r=1 was treated, and taken together, our results imply that R has finite F-representation type (FFRT). Though it is expected that all rings of invariants for reductive groups have FFRT, ours is the first non-trivial example of such a ring for a group which is not linearly reductive. As a corollary, we show that the ring of differential operators Dk(R) is simple, that G has global finite F-representation type (GFFRT) and that R provides a noncommutative resolution for Rpr.

Finite Generation of Cox Rings

In this expository note we discuss a class of graded algebras named Cox rings, which are naturally associated to algebraic varieties generalizing the homogeneous coordinate rings of projective spaces. Whenever the Cox ring is finitely generated, the variety admits a quotient presentation by a quasitorus, which resembles the quotient construction of the projective space. We discuss the problem of the finite generation of Cox rings from a geometric perspective and provide examples of both the finitely and non-finitely generated cases.

Reduced arc schemes for Veronese embeddings and global Demazure modules

We consider arc spaces for the compositions of Plücker and Veronese embeddings of the flag varieties for simple Lie groups of types ADE. The arc spaces are not reduced and we consider the homogeneous coordinate rings of the corresponding reduced schemes. We show that each graded component of a homogeneous coordinate ring is a cocyclic module over the current algebra and is acted upon by the algebra of symmetric polynomials. We show that the action of the polynomial algebra is free and that the fiber at the special point of a graded component is isomorphic to an affine Demazure module whose level is the degree of the Veronese embedding. In type A1 (which corresponds to the Veronese curve), we give the precise list of generators of the reduced arc space. In general type, we introduce the notion of global higher level Demazure modules, which generalizes the standard notion of the global Weyl modules, and identify the graded components of the homogeneous coordinate rings with these modules.

A conjecture strengthening the Zariski dense orbit problem for birational maps of dynamical degree one

AbstractWe formulate a strengthening of the Zariski dense orbit conjecture for birational maps of dynamical degree one. So, given a quasiprojective variety X defined over an algebraically closed field K of characteristic $0$ , endowed with a birational self-map $\phi $ of dynamical degree $1$ , we expect that either there exists a nonconstant rational function $f:X\dashrightarrow \mathbb {P} ^1$ such that $f\circ \phi =f$ , or there exists a proper subvariety $Y\subset X$ with the property that, for any invariant proper subvariety $Z\subset X$ , we have that $Z\subseteq Y$ . We prove our conjecture for automorphisms $\phi $ of dynamical degree $1$ of semiabelian varieties X. Moreover, we prove a related result for regular dominant self-maps $\phi $ of semiabelian varieties X: assuming that $\phi $ does not preserve a nonconstant rational function, we have that the dynamical degree of $\phi $ is larger than $1$ if and only if the union of all $\phi $ -invariant proper subvarieties of X is Zariski dense. We give applications of our results to representation-theoretic questions about twisted homogeneous coordinate rings associated with abelian varieties.

Theta functions on varieties with effective anti-canonical class

We show that a large class of maximally degenerating families of n n -dimensional polarized varieties comes with a canonical basis of sections of powers of the ample line bundle. The families considered are obtained by smoothing a reducible union of toric varieties governed by a wall structure on a real n n -(pseudo-)manifold. Wall structures have previously been constructed inductively for cases with locally rigid singularities [Gross and Siebert, From real affine geometry to complex geometry (2011)] and by Gromov-Witten theory for mirrors of log Calabi-Yau surfaces and K 3 K3 surfaces [Gross, Pandharipande and Siebert, The tropical vertex; Gross, Hacking and Keel, Mirror symmetry for log Calabi-Yau surfaces (2015); Gross, Hacking, Keel, and Siebert, Theta functions and K 3 K3 surfaces (In preparation)]. For trivial wall structures on the n n -torus we retrieve the classical theta functions. We anticipate that wall structures can be constructed quite generally from maximal degenerations. The construction given here then provides the homogeneous coordinate ring of the mirror degeneration along with a canonical basis. The appearance of a canonical basis of sections for certain degenerations points towards a good compactification of moduli of certain polarized varieties via stable pairs, generalizing the picture for K3 surfaces [Gross, Hacking, Keel, and Siebert, Theta functions and K 3 K3 surfaces (In preparation)]. Another possible application apart from mirror symmetry may be to geometric quantization of varieties with effective anti-canonical class.

Beyond the Sottile–Sturmfels Degeneration of a Semi-Infinite Grassmannian

Abstract We study toric degenerations of semi-infinite Grassmannians (a.k.a. quantum Grassmannians). While the toric degenerations of the classical Grassmannians are well studied, the only known example in the semi-infinite case is due to Sottile and Sturmfels. We start by providing a new interpretation of the Sottile–Sturmfels construction by finding a poset such that their degeneration is the toric variety of the order polytope of the poset. We then use our poset to construct and study a new toric degeneration in the semi-infinite case. Our construction is based on the notion of poset polytopes introduced by Fang–Fourier–Litza–Pegel. As an application, we introduce semi-infinite PBW-semistandard tableaux, giving a basis in the homogeneous coordinate ring of a semi-infinite Grassmannian.

Brane Quantization of Toric Poisson Varieties

In this paper we propose a noncommutative generalization of the relationship between compact K\"ahler manifolds and complex projective algebraic varieties. Beginning with a prequantized K\"ahler structure, we use a holomorphic Poisson tensor to deform the underlying complex structure into a generalized complex structure, such that the prequantum line bundle and its tensor powers deform to a sequence of generalized complex branes. Taking homomorphisms between the resulting branes, we obtain a noncommutative deformation of the homogeneous coordinate ring. As a proof of concept, this is implemented for all compact toric K\"ahler manifolds equipped with an R-matrix holomorphic Poisson structure, resulting in what could be called noncommutative toric varieties. To define the homomorphisms between generalized complex branes, we propose a method which involves lifting each pair of generalized complex branes to a single coisotropic A-brane in the real symplectic groupoid of the underlying Poisson structure, and compute morphisms in the A-model between the Lagrangian identity bisection and the lifted coisotropic brane. This is done with the use of a multiplicative holomorphic Lagrangian polarization of the groupoid.

The Canonical Module of GT-Varieties and the Normal Bundle of RL-Varieties

In this paper, we study the geometry of GT-varieties X_{d} with group a finite cyclic group Gamma subset {{,mathrm{GL},}}(n+1,mathbb {K}) of order d. We prove that the homogeneous ideal {{,mathrm{I},}}(X_{d}) of X_{d} is generated by binomials of degree at most 3 and we provide examples reaching this bound. We give a combinatorial description of the canonical module of the homogeneous coordinate ring of X_{d} and we show that it is generated by monomial invariants of Gamma of degree d and 2d. This allows us to characterize the Castelnuovo–Mumford regularity of the homogeneous coordinate ring of X_d. Finally, we compute the cohomology table of the normal bundle of the so-called RL-varieties. They are projections of the Veronese variety nu _{d}(mathbb {P}^{n}) subset mathbb {P}^{left( {begin{array}{c}n+d nend{array}}right) -1} which naturally arise from level GT-varieties.

Mirror symmetry for the Tate curve via tropical and log corals

We introduce tropical corals, balanced trees in a half-space, and show that they correspond to holomorphic polygons capturing the product rule in Lagrangian Floer theory for the elliptic curve. We then prove a correspondence theorem equating counts of tropical corals to punctured log Gromov–Witten invariants of the Tate curve. This implies that the homogeneous coordinate ring of the mirror to the Tate curve is isomorphic to the degree-zero part of symplectic cohomology, confirming a prediction of homological mirror symmetry.

Calabi–Yau properties of Postnikov diagrams

Abstract We show that the dimer algebra of a connected Postnikov diagram in the disc is bimodule internally $3$ -Calabi–Yau in the sense of the author’s earlier work [43]. As a consequence, we obtain an additive categorification of the cluster algebra associated to the diagram, which (after inverting frozen variables) is isomorphic to the homogeneous coordinate ring of a positroid variety in the Grassmannian by a recent result of Galashin and Lam [18]. We show that our categorification can be realised as a full extension closed subcategory of Jensen–King–Su’s Grassmannian cluster category [28], in a way compatible with their bijection between rank $1$ modules and Plücker coordinates.

A symplectic look at the Fargues–Fontaine curve

Abstract We study a version of the Fukaya category of a symplectic 2-torus with coefficients in a locally constant sheaf of rings. The sheaf of rings includes a globally defined Novikov parameter that plays its usual role in organising polygon counts by area. It also includes a ring of constants whose variation around the the torus can be encoded by a pair of commuting ring automorphisms. When these constants are perfectoid of characteristic p, one of the holonomies is trivial and the other is the $p^{th}$ power map, it is possible in a limited way to specialise the Novikov parameter to 1. We prove that the Dehn twist ring defined there is isomorphic to the homogeneous coordinate ring of a scheme introduced by Fargues and Fontaine: their ‘curve of p-adic Hodge theory’ for the local field $\mathbf {F}_p(\!(z)\!)$ .

Presentations of the Roger–Yang generalized skein algebra

We describe presentations of the Roger-Yang generalized skein algebras for punctured spheres with an arbitrary number of punctures. This skein algebra is a quantization of the decorated Teichmuller space and generalizes the construction of the Kauffman bracket skein algebra. In this paper, we also obtain a new interpretation of the homogeneous coordinate ring of the Grassmannian of planes in terms of skein theory.

Identifying limits of ideals of points in the case of projective space

We study the closure of the locus of radical ideals in the multigraded Hilbert scheme associated with a standard graded polynomial ring and the Hilbert function of a homogeneous coordinate ring of points in general position in projective space. In the case of projective plane, we give a sufficient condition for an ideal to be in the closure of the locus of radical ideals. For projective space of arbitrary dimension we present a necessary condition. The paper is motivated by the border apolarity lemma which connects such multigraded Hilbert schemes with the theory of ranks of polynomials.