Affine Cone Research Articles

Dynamics of Chebyshev endomorphisms on some affine algebraic varieties

The Chebyshev polynomials Td in one variable are typical chaotic maps on C. Chebyshev endomorphisms PAnd: Cn→Cn are also chaotic. We consider the action of the dihedral group Dn+1 on Cn. The endomorphism PAnd maps any Dn+1-orbit of z∈Cn to a Dn+1-orbit of PAnd(z). The endomorphism PAnd induces a mapping on Cn∕Dn+1. Using invariant theory, we embed Cn∕Dn+1 as an affine subvariety X in Cm. Then we have morphisms gd on X. We study the cases n=2 and 3. In these cases, the morphisms gd are defined over Z. We find a class of affine subvarieties V of X which are invariant under gd. These varieties are concerned with branch loci or critical loci. The class contains C2, a cuspidal cubic, a parabola, a quadric hypersurface in C4, an affine algebraic surface in C4 which is birationally equivalent to an affine quadric cone in C3, and others. For each affine variety V in the class, there exists a polynomial parameterization PV satisfying gd|V(PV(y1,…,yk))=PV(Td(y1),…,Td(yk)), where Td(z) is a Chebyshev polynomial in one variable. Then we determine the set of bounded orbits of gd|V in each invariant set V and give relations between them.

A note on affine cones over Grassmannians and their stringy 𝐸-functions

We compute the stringy E E -function of the affine cone over a Grassmannian. If the Grassmannian is not a projective space then its cone does not admit a crepant resolution. Nonetheless the stringy E E -function is sometimes a polynomial and in those cases the cone admits a noncommutative crepant resolution. This raises the question as to whether the existence of a noncommutative crepant resolution implies that the stringy E E -function is a polynomial.

On a lower bound of de Rham depth of affine cone of a projective space in characteristic zero

Let be a proper closed subset in the projective n-space over a field k of characteristic zero and let C(Y) be its affine cone. Let be the homogeneous defining ideal of Y with homogeneous maximal ideal . In this paper, we find out a lower bound of de Rham depth of C(Y) in terms of .

Locally Nilpotent Derivations of Graded Integral Domains and Cylindricity

Let B be a commutative \(\mathbb {Z}\)-graded domain of characteristic zero. An element f of B is said to be cylindrical if it is nonzero, homogeneous of nonzero degree, and such that B(f) is a polynomial ring in one variable over a subring. We study the relation between the existence of a cylindrical element of B and the existence of a nonzero locally nilpotent derivation of B. Also, given d ≥ 1, we give sufficient conditions that guarantee that every derivation of \(B^{(d)} = {\bigoplus }_{i \in \mathbb {Z}} B_{di}\) can be extended to a derivation of B. We generalize some results of Kishimoto, Prokhorov and Zaidenberg that relate the cylindricity of a polarized projective variety (Y,H) to the existence of a nontrivial Ga-action on the affine cone over (Y,H).

On regularity of symbolic Rees algebras and symbolic powers of vertex cover ideals of graphs

In this work, we study the bigraded regularities of the symbolic Rees algebras R s ( J ( G ) ) , R s ( I ( G ) ) R_s(J(G)), R_s(I(G)) , of the vertex cover ideal J ( G ) J(G) and the edge ideal I ( G ) I(G) , of a graph G G respectively. We give combinatorial upper bounds for the ( 1 , 0 ) (1,0) -regularities of R s ( J ( G ) ) R_s(J(G)) and R s ( I ( G ) ) R_s(I(G)) . By using this upper bounds, we give general linear upper bounds for r e g ( J ( G ) ( k ) ) , r e g ( I ( G ) ( k ) ) reg(J(G)^{(k)}), reg(I(G)^{(k)}) for any k ≥ 1 k\geq 1 . Let G G be a graph on n n vertices and deg ⁡ ( J ( G ) ) \deg (J(G)) be the maximum degree of minimal generators of J ( G ) J(G) . We show that if G G is a non-bipartite graph, then k deg ⁡ ( J ( G ) ) ≤ r e g ( J ( G ) ( k ) ) ≤ k deg ⁡ ( J ( G ) ) + α 0 ( G ) − 1 + | A 0 ∪ { x i 1 , … , x i r } | − r , \begin{equation*} k \deg (J(G)) \!\leq \! reg(J(G)^{(k)})\!\leq \! k \deg (J(G))+ \alpha _0(G)-1+|A_0\cup \{x_{i_1}, \ldots , x_{i_r}\}|-r, \end{equation*} for all k ≥ 1 k \geq 1 , where α 0 ( G ) \alpha _0(G) denotes the vertex cover number of G G , A 0 A_0 is a maximal independent set in G G of maximal cardinality, and r r is the number of 0 0 -covers that are present in an irreducible representation of the affine cone associated with the irreducible covers of G G . Also if G G is a non-bipartite perfect graph, then 2 k ≤ r e g ( I ( G ) ( k ) ) ≤ 2 k + n − r + 1 , \begin{equation*} 2k \leq reg(I(G)^{(k)})\leq 2k+n-r+1, \end{equation*} for all k ≥ 1 k \geq 1 , where r r is the number of 0 0 -covers of Γ ( G ) \Gamma (G) that are present in an irreducible representation of the affine cone associated with the irreducible covers of Γ ( G ) \Gamma (G) .

Weighted K-stability of polarized varieties and extremality of Sasaki manifolds

We use the correspondence between extremal Sasaki structures and weighted extremal Kähler metrics defined on a regular quotient of a Sasaki manifold, established by the first two authors in [1], and the theory of weighted K-stability (introduced by Lahdili in [39]) in order to define a suitable notion of (relative) weighted K-stability for compact Sasaki manifolds of regular type. We show that the (relative) weighted K-stability with respect to a maximal torus and smooth equivariant test configurations with reduced central fibre is a necessary condition for the existence of a (possibly irregular) extremal Sasaki metric. We also compare weighted K-stability to the K-stability of the corresponding polarized affine cone (introduced by Collins–Székelyhidi in [18]), and prove that they agree on the class of test configurations we consider. As a byproduct, we strengthen the obstruction to the existence of a scalar-flat Kähler cone metric found in [18] from the K-semistability to the K-stability on these test configurations. We use our approach to give a characterization of the existence of a compatible extremal Sasaki structure on a principal S1-bundle over an admissible ruled manifold in the sense of [4], expressed in terms of the positivity of a single polynomial of one variable over a given interval.

Affine cones over cubic surfaces are flexible in codimension one

Abstract Let Y be a smooth del Pezzo surface of degree 3 polarized by a very ample divisor that is not proportional to the anticanonical one. Then the affine cone over Y is flexible in codimension one. Equivalently, such a cone has an open subset with an infinitely transitive action of the special automorphism group on it.

Tropical geometry and Newton–Okounkov cones for Grassmannian of planes from compactifications

AbstractWe construct a family of compactifications of the affine cone of the Grassmannian variety of $2$ -planes. We show that both the tropical variety of the Plücker ideal and familiar valuations associated to the construction of Newton–Okounkov bodies for the Grassmannian variety can be recovered from these compactifications. In this way, we unite various perspectives for constructing toric degenerations of flag varieties.

On sharp rates and analytic compactifications of asymptotically conical Kähler metrics

Let X be a complex manifold, and let S↪X be an embedding of a complex submanifold. Assuming that the embedding is (k−1)-linearizable or (k−1)-comfortably embedded, we construct via the deformation to the normal cone a diffeomorphism F from a small neighborhood of the zero section in the normal bundle NS to a small neighborhood of S in X such that F is in a precise sense holomorphic up to the (k−1)th order. Using this F, we obtain optimal estimates on asymptotic rates for asymptotically conical (AC) Calabi–Yau (CY) metrics constructed by Tian and Yau. Furthermore, when S is an ample divisor satisfying an appropriate cohomological condition, we relate the order of comfortable embedding to the weight of the deformation of the normal isolated cone singularity arising from the deformation to the normal cone. We also give an example showing that the condition of comfortable embedding depends on the splitting liftings. We then prove an analytic compactification result for the deformation of the complex structure on an affine cone that decays to any positive order at infinity. This can be seen as an analytic counterpart of Pinkham’s result on deformations of cone singularities with negative weights.

The monic rank

We introduce the monic rank of a vector relative to an affine-hyperplane section of an irreducible Zariski-closed affine cone $X$. We show that the monic rank is finite and greater than or equal to the usual $X$-rank. We describe an algorithmic technique based on classical invariant theory to determine, in concrete situations, the maximal monic rank. Using this technique, we establish three new instances of a conjecture due to B. Shapiro which states that a binary form of degree $d\cdot e$ is the sum of $d$ $d$-th powers of forms of degree $e$. Furthermore, in the case where $X$ is the cone of highest weight vectors in an irreducible representation---this includes the well-known cases of tensor rank and symmetric rank---we raise the question whether the maximal rank equals the maximal monic rank. We answer this question affirmatively in several instances.

The uniform symbolic topology property for diagonally F-regular algebras

Let k be a field of positive characteristic. Building on the work of the second named author, we define a new class of k-algebras, called diagonally F-regular algebras, for which the so-called Uniform Symbolic Topology Property (USTP) holds effectively. We show that this class contains all essentially smooth k-algebras. We also show that this class contains certain singular algebras, such as the affine cone over Pkr×Pks, when k is perfect. By reduction to positive characteristic, it follows that USTP holds effectively for the affine cone over PCr×PCs and more generally for complex varieties of diagonal F-regular type.

Failure of strong approximation on an affine cone

We use the Brauer-Manin obstruction to strong approximation on a punctured affine cone to explain a curious property of coprime integer solutions to a homogeneous Diophantine equation.

Kähler–Einstein metrics and volume minimization

We prove that if a polarized Q-Fano variety admits a Kähler–Einstein metric, then the normalized volume functional over the associated affine cone is globally minimized at the canonical divisorial valuation obtained by blowing-up the vertex. This is also generalized to the logarithmic and the orbifold settings. A refinement of this result together with the result in [41] gives an equivalent characterization of K-semistability for any smooth Fano manifold. We also prove that the valuation associated to the Reeb vector field of a smooth Sasaki–Einstein metric minimizes the normalized volume over the corresponding Kähler cone. These results strengthen and generalize the minimization result of Martelli–Sparks–Yau.

Geometric classification of 4d mathcal{N}=2 SCFTs

The classification of 4d mathcal{N}=2 SCFTs boils down to the classification of conical special geometries with closed Reeb orbits (CSG). Under mild assumptions, one shows that the underlying complex space of a CSG is (birational to) an affine cone over a simply-connected ℚ-factorial log-Fano variety with Hodge numbers hp,q = δp,q. With some plausible restrictions, this means that the Coulomb branch chiral ring is a graded polynomial ring generated by global holomorphic functions ui of dimension Δi. The coarse-grained classification of the CSG consists in listing the (finitely many) dimension k-tuples {Δ1, Δ2, ⋯ , Δk} which are realized as Coulomb branch dimensions of some rank-k CSG: this is the problem we address in this paper. Our sheaf-theoretical analysis leads to an Universal Dimension Formula for the possible {Δ1, ⋯ , Δk}’s. For Lagrangian SCFTs the Universal Formula reduces to the fundamental theorem of Springer Theory.The number N(k) of dimensions allowed in rank k is given by a certain sum of the Erdös-Bateman Number-Theoretic function (sequence A070243 in OEIS) so that for large kNk=2ζ2ζ3ζ6k2+ok2.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\boldsymbol{N}(k)=\\frac{2\\zeta (2)\\zeta (3)}{\\zeta (6)}{k}^2+o\\left({k}^2\\right). $$\\end{document}In the special case k = 2 our dimension formula reproduces a recent result by Argyres et al.Class Field Theory implies a subtlety: certain dimension k-tuples {Δ1, ⋯ , Δk} are consistent only if supplemented by additional selection rules on the electro-magnetic charges, that is, for a SCFT with these Coulomb dimensions not all charges/fluxes consistent with Dirac quantization are permitted.Since the arguments tend to be abstract, we illustrate the various aspects with several concrete examples and perform a number of explicit checks. We include detailed tables of dimensions for the first few k’s.

Connectedness and Lyubeznik Numbers

AbstractWe investigate the relationship between connectedness properties of spectra and the Lyubeznik numbers, numerical invariants defined via local cohomology. We prove that for complete equidimensional local rings, the Lyubeznik numbers characterize when connectedness dimension equals 1. More generally, these invariants determine a bound on connectedness dimension. Additionally, our methods imply that the Lyubeznik number $\lambda _{1,2}(A)$ of the local ring $A$ at the vertex of the affine cone over a projective variety is independent of the choice of its embedding into projective space.

Fano–Mukai fourfolds of genus 10 as compactifications of $${\mathbb {C}}^4$$ C 4

It is known that the moduli space of smooth Fano–Mukai fourfolds $$V_{18}$$ of genus 10 has dimension one. We show that any such fourfold is a completion of $${\mathbb {C}}^4$$ in two different ways. Up to isomorphism, there is a unique fourfold $$V_{18}^{{\mathrm {s}}}$$ acted upon by $${\mathrm{SL}}_2({\mathbb {C}})$$ . The group is a semidirect product . Furthermore, $$V_{18}^{{\mathrm {s}}}$$ is a $${\mathrm{GL}}_2({\mathbb {C}})$$ -equivariant completion of $${\mathbb {C}}^4$$ , and as well of $${\mathrm{GL}}_2({\mathbb {C}})$$ . The restriction of the $${\mathrm{GL}}_2({\mathbb {C}})$$ -action on $$V_{18}^{{\mathrm {s}}}$$ to yields a faithful representation with an open orbit. There is also a unique, up to isomorphism, fourfold $$V_{18}^{\mathrm a}$$ such that the group is a semidirect product . For a Fano–Mukai fourfold $$V_{18}$$ isomorphic neither to $$V_{18}^{{\mathrm {s}}}$$ , nor to $$V_{18}^{\mathrm a}$$ , the group is a semidirect product of $$({{\mathbb {G}}}_{\mathrm {m}})^2$$ and a finite cyclic group whose order is a factor of 6. Besides, we establish that the affine cone over any polarized Fano–Mukai variety $$V_{18}$$ is flexible in codimension one, and flexible if $$V_{18}=V_{18}^{{\mathrm {s}}}$$ .

The volume of singular Kähler–Einstein Fano varieties

We show that the anti-canonical volume of an$n$-dimensional Kähler–Einstein$\mathbb{Q}$-Fano variety is bounded from above by certain invariants of the local singularities, namely$\operatorname{lct}^{n}\cdot \operatorname{mult}$for ideals and the normalized volume function for real valuations. This refines a recent result by Fujita. As an application, we get sharp volume upper bounds for Kähler–Einstein Fano varieties with quotient singularities. Based on very recent results by Li and the author, we show that a Fano manifold is K-semistable if and only if a de Fernex–Ein–Mustaţă type inequality holds on its affine cone.

K3 transitions and canonical 3-folds

We introduce K3 transitions as a geometric approach to studying canonical 3-folds. These transitions link different deformation classes of canonical 3-folds via a combination of birational contractions and smoothings. As applications, we investigate some basic properties of the web of canonical 3-folds in small codimension and give an interesting example of a singularity in codimension 4 with obstructed smoothings, similar to the famous example of the affine cone over a degree 6 del Pezzo surface.

K-theoretic Defect in Chern Class Identity for a Free Divisor

Let $X$ be a nonsingular variety defined over an algebraically closed field of characteristic $0$, and $D$ be a free divisor. We study the motivic Chern class of $D$ in the Grothendieck group of coherent sheaves $G_0(X)$, and another class defined by the sheaf of logarithmic differentials along $D$. We give explicit calculations of the difference of these two classes when: $D$ is a divisor on a nonsingular surface; $D$ is a hyperplane arrangement whose affine cone is free.

Hodge theory and deformations of affine cones of subcanonical projective varieties

We investigate the relation between the Hodge theory of a smooth subcanonical $n$-dimensional projective variety $X$ and the deformation theory of the affine cone $A_X$ over $X$. We start by identifying $H^{n-1,1}_{\mathrm{prim}}(X)$ as a distinguished graded component of the module of first order deformations of $A_X$, and later on we show how to identify the whole primitive cohomology of $X$ as a distinguished graded component of the Hochschild cohomology module of the punctured affine cone over $X$. In the particular case of a projective smooth hypersurface $X$ we recover Griffiths' isomorphism between the primitive cohomology of $X$ and certain distinguished graded components of the Milnor algebra of a polynomial defining $X$. The main result of the article can be effectively exploited to compute Hodge numbers of smooth subcanonical projective varieties. We provide a few example computation, as well a SINGULAR code, for Fano and Calabi-Yau threefolds.