This is a survey of our research on geometric structures of projective embeddings and includes some topics of our talks in several symposia during 1990-99. We clarify our main problem, which is to construct a kind of geometric composition series of projective embeddings. The concept of "geometric composition series" is an analogy in Algebraic Geometry with Jordan-Hölder series in Group Theory. We present two of the candidates for the construction problem. To approach this problem, we show several results and new tools for handling higher obstructions appearing in infinitesimal liftings. As a byproduct of the tools, we obtain a simplified proof for a criterion on arithmetic normality described in terms of Differential Geometry. (Keywords): Petri's analysis, Second fundamental form, Syzygy, Arithmetic normality, Infinitesimal lifting, Geometric shell, Geometric composition series, Lefschetz chain, Dual Lefschetz chain, Meta-Lefschetz operator.
We prove an elementary but somewhat unexpected result about projective embeddings of smooth varieties X whose cotangent bundles are numerically effective. Specifically, we show that the degree of X in any projective embedding must grow (almost) exponentially in the dimension of X.
We construct complex projective schemes with Lyubeznik numbers of their cones depending on the choices of projective embeddings. This answers a question of G. Lyubeznik in the characteristic 0 case. It contrasts with a theorem of W. Zhang in the positive characteristic case where the Frobenius endomorphism is used. Reducibility of schemes is essential in our argument. B. Wang recently constructed examples of irreducible projective schemes (which are not normal) from our examples of reducible ones. So the question is still open in the normal singular case.
Let $\hat{L}$ be the projective completion of an ample line bundle $L$ over $D$, a smooth projective manifold. Hwang-Singer \cite{HwangS} have constructed complete CSCK metric on $\hat{L}\backslash D$. When the corresponding \kahler form is in the cohomology class of a rational divisor $A$ and when $L$ has negative CSCK metric on $D$, we show that the Kodaira embedding induced by orthonormal basis of the Bergman space of $kA$ is almost balanced. As a corollary, $(\hat{L},D,cA,0)$ is K-semistable.
We embed polarised orbifolds with cyclic stabiliser groups into weighted projective space via a weighted form of Kodaira embedding. Dividing by the (non-reductive) automorphisms of weighted projective space then formally gives a moduli space of orbifolds. We show how to express this as a reductive quotient and so a GIT problem, thus defining a notion of stability for orbifolds. We then prove an orbifold version of Donaldson's theorem: the existence of an orbifold Kahler metric of constant scalar curvature implies K-semistability. By extending the notion of slope stability to orbifolds we there…
Geometric structures modeled on rational homogeneous manifolds are studied to characterize rational homogeneous manifolds and to prove their deformation rigidity. To generalize these characterizations and deformation rigidity results to quasihomogeneous varieties, we first study horospherical varieties and geometric structures modeled on horospherical varieties. Using Cartan geometry, we prove that a geometric structure modeled on a smooth projective horospherical variety of Picard number one is locally equivalent to the standard geometric structure when the geometric structure is defined on a…
An explicit projective embedding of the moduli space of marked cubic surfaces is given. This embedding is equivariant under the Weyl group of type E6. The image is defined by a system of linear and cubic equations. To express the embedding in a most symmetric way, the target would be 79-dimensional, however the image lies in a 9-dimensional linear subspace.
An important result of Zhang states that for a projective variety, the existence of a balanced embedding is equivalent to Chow stability. In this paper, we shall prove that Chow stability implies that a balanced embedding exists via the continuity method. Our proof is conditional on a technical hypothesis about restrictions of Hamiltonians to subschemes of projective space.
Given a smooth polarized Riemann surface (X, L) endowed with a hyperbolic metric $ω$ with cusp singularities along a divisor D, we show the L^2 projective embedding of (X, D) defined by L^k is asymptotically almost balanced in a weighted sense. The proof depends on sufficiently precise understanding of the behavior of the Bergman kernel in three regions, with the most crucial one being the neck region around D. This is the first step towards understanding the algebro-geometric stability of extremal Kähler metrics with singularities.
We construct irreducible complex projective varieties such that the Lyubeznik numbers of their affine cones depend on the choices of projective embeddings. The main ingredient is the recent work of Reichelt-Saito-Walther, where the Lyubeznik numbers are reinterpreted using perverse sheaves and reducible projective varieties with dependent Lyubeznik numbers are constructed.
A criterion for the existence of a birational embedding of an algebraic curve into a projective plane with two Galois points is presented. Several novel examples of plane curves with two inner Galois points as an application are described.
This paper deals with syzygies of Segre embeddings. Let d >=3 and n_1, ..., n_d nonzero natural numbers. We prove that O(1,...., 1) on the product of P^{n_1}, ...,P^{n_d} satisfies Property N_p if and only if p <= 3.
Let $C \subset \mathbb {P}^r$ a general embedding of prescribed degree of a general smooth curve with prescribed genus. Here we prove that either $h^0(\mathbb {P}^r,\mathcal {I}_C(2)) =0$ or $h^1(\mathbb {P}^r,\mathcal {I}_C(2)) =0$ (a problem called the Maximal Rank Conjecture in the range of quadrics).
Let X be a nonsingular arithmetically Cohen-Macaulay projective scheme, Z a nonsingular subscheme of X. Let π: Y --> X be the blowup of X along the ideal sheaf of Z, E_0 the pull-back of a general hyperplane in X and E the exceptional divisor. In this paper, we study projective embeddings of Y given by the divisor tE_0 - eE. We give explicit values of d and δsuch that for all e > 0 and t > ed + δ, these embeddings is projectively normal and arithmetically Cohen-Macaulay. We also study the regularity of the ideal sheaf and syzygies of these embeddings. When X is a surface and Z is a 0-dimension…
We describe explicitly how certain standard opens of the Hilbert scheme of points are embedded into Grassmannians. The standard opens of the Hilbert scheme that we consider are given as the intersection of a corresponding basic open affine of the Grassmannian and a closed stratum determined by a Fitting ideal.
We study embeddability of superschemes into $Π$-projective spaces and into supergrassmannians $G(1|1,n|n)$. We give some criteria based on the relation with ${\mathbb{A}}^{0|1}$-torsors and ${\mathbb{A}}^{0|1}$-fibrations. We also prove the existence of nice quotients for free actions of ${\mathbb{A}}^{0|1}$ on superschemes.
We study open equivariant projective embeddings of homogeneous spaces such that the complement of the open orbit does not contain divisors. Criterions of existence of such an embedding are considered and finiteness of isomorphism classes of embeddings for a given homogeneous space is proved. Any embedding with small boundary is realized as a GIT-quotient associated with a linearization of the trivial line bundle on the space of the canonical embedding. The generalized Cox's construction and the theory of bunched rings allow us to describe basic geometric properties of embeddings with small bou…
Let V be a normal affine surface which admits a C*- and a C+-action. In this note we show that in many cases V can be embedded as a principal Zariski open subset into a hypersurface of a weighted projective space. In particular, we recover a result of D. Daigle and P. Russell.
In this paper, we prove that any two birational projective varieties with finite quotient singularities can be realized as two geometric GIT quotients of a non-singular projective variety by a reductive algebraic group. Then, by applying the theory of Variation of Geometric Invariant Theory Quotients ([3]), we show that they are related by a sequence of GIT wall-crossing flips.
We provide a geometric approach to constructing Lefschetz collections and Landau-Ginzburg Homological Projective Duals from a variation of Geometric Invariant Theory quotients. This approach yields homological projective duals for Veronese embeddings in the setting of Landau Ginzburg models. Our results also extend to a relative Homological Projective Duality framework.
The projective line over a field carries structure of a groupoid with a certain correspondence between objects and arrows. We discuss to what extent the field can be reconstructed from the groupoid.
Given an embedding of a smooth projective curve $X$ of genus $g\geq1$ into $\mathbb{P}^N$, we study the locus of linear subspaces of $\mathbb{P}^N$ of codimension 2 such that projection from said subspace, composed with the embedding, gives a Galois morphism $X\to\mathbb{P}^1$. For genus $g\geq2$ we prove that this locus is a smooth projective variety with components isomorphic to projective spaces. If $g=1$ and the embedding is given by a complete linear system, we prove that this locus is also a smooth projective variety whose positive-dimensional components are isomorphic to projective bund…
An embedding of a point-line geometry Γis usually defined as an injective mapping εfrom the point-set of Γto the set of points of a projective space such that ε(l) is a projective line for every line l of Γ, but different situations have lately been considered in the literature, where ε(l) is allowed to be a subline of a projective line or a curve. In this paper we propose a more general definition of embedding which includes all the above situations and we focus on a class of embeddings, which we call Grassmman embeddings, where the points of Γare firstly associated to lines of a projective g…
The Deligne-Lusztig varieties associated to the Coxeter classes of the algebraic groups 2A2, 2B2 and 2G2 are affine algebraic curves. We produce explicit projective models of the closures of these curves. Furthermore for $d$ the Coxeter number of these groups, we find polynomials for each of these models that cut out the $\F_q$-points, the $\F_{q^d}$-points and the $\F_{q^{d+1}}$-points, and demonstrate a relation satisfied by these polynomials.
We show that if PGA is understood as a subalgebra of CGA in mathematically correct sense, then the flat objects share the same representation in PGA and CGA. Particularly, we treat duality in PGA. This leads to unification of PGA and CGA objects which is important especially for software implementation and symbolic calculations.
In this article we generalize the injectivity of the push-forward homomorphism at the level of Chow groups, induced by the closed embedding of $\Sym^m C$ into $\Sym^n C$ for $m\leq n$, where $C$ is a smooth projective curve, to symmetric powers of a smooth projective variety of higher dimension. We also prove the analog of this theorem for product of symmetric powers of smooth projective varieties. As an application we prove the injectivity of the push-forward homomorphism at the level of Chow groups, induced by the closed embedding of self product of theta divisor into the self product of the…
This is an expository paper in which we define projective GIT quotients and introduce toric varieties from this perspective. It is intended primarily for readers who are learning either invariant theory or toric geometry for the first time.
Our primary result is that a demi-normal quasi-projective variety can be embedded in a demi-normal projective variety. Recall that a demi-normal variety $X$ is a variety with properties $S_2$, $G_1$, and seminormality. Equivalently, $X$ has Serre's $S_2$ property and there is an open subvariety $U$ with complement of codimension at least 2 in $X$, such that the only singularities of $U$ are (analytically) double normal crossings. The term demi-normal was coined by Kollár in \cite{Kol13}. As a consequence of this embedding theorem, we prove a semi-smooth Grauert-Riemenschneider vanishing theore…
In this paper we study pluricanonical maps of minimal projective 3-folds of general type with geometric genus $1$, $2$ and $3$. We go in the direction pioneered by Enriques and Bombieri, and other authors, pinning down, for low projective genus, a finite list of exceptions to the birationality of some pluricanonical map. In particular, apart from a finite list of weighted baskets, we prove the birationality of $φ_{16}$, $φ_{6}$ and $φ_{5}$ respectively.
The notion of higher order dual varieties of a projective variety is a natural generalization of the classical notion of projective duality, introduced by Piene in 1983. In this paper we study higher order dual varieties of projective toric embeddings. We compute the degree of the second dual variety of a smooth toric threefold in geometric and combinatorial terms, and we classify smooth 2-jet spanned projective embeddings of smooth threefolds whose second dual variety has dimension less than expected. We also describe the tropicalization of the k-th dual variety of an equivariantly embedded (…
This paper is devoted to extend some Hu-Keel results on Mori dream spaces (MDS) beyond the projective setup. Namely, $\Q$-factorial algebraic varieties with finitely generated class group and Cox ring, here called \emph{weak} Mori dream spaces (wMDS), are considered. Conditions guaranteeing the existence of a neat embedding of a (completion of a) wMDS into a complete toric variety are studied, showing that, on the one hand, those which are complete and admitting low Picard number are always projective, hence Mori dream spaces in the sense of Hu-Keel. On the other hand, an example of a wMDS doe…
University of Ibadan, Faculty of Science（伊巴丹大学理学院）；Technische Universität Berlin, Institut für Mathematik（柏林工业大学数学研究所）；Universität Wien, Fakultät für Mathematik（维也纳大学数学系）
We describe the Kuznetsov component of the Plücker embedding of the Grassmannian as a category of matrix factorizations on an noncommutative crepant resolution (NCCR) of the affine cone of the Grassmannian. We also extend this to a full homological projective dual (HPD) statement for the Plücker embedding. The first part is finding and describing the NCCR, which is also of independent interest. We extend results of Špenko and Van den Bergh to prove the existence of an NCCR for the affine cone of the Grassmannian. We then relate this NCCR to a categorical resolution of Kuznetsov. Deforming thes…
We give a criterion for a projective surface to become a quotient of a fake projective plane. We also give a detailed information on the elliptic fibration of a $(2,3)$-elliptic surface that is the minimal resolution of a quotient of a fake projective plane. As a consequence, we give a classification of ${\mathbb Q}$-homology projective planes with cusps only.
We characterize $d$-uple Veronese embeddings of finite-dimensional projective spaces. The easiest non-trivial instance of our theorem is the embedding of the projective plane in 5-dimensional projective space, a result obtained in 1901 by Severi when the underlying field is complex.
Fake projective planes are smooth complex surfaces of general type with Betti numbers equal to that of the usual projective plane. Recent explicit constructions of fake projective planes embed them via their bicanonical embedding in $\mathbb P^9$. In this paper, we study Keum's fake projective plane $(a=7, p=2, \{7\}, D_3 2_7)$ and use the equations of \cite{Borisov} to construct an embedding of fake projective plane in $\mathbb P^5$. We also simplify the 84 cubic equations defining the fake projective plane in $\mathbb P^9$.
The moduli space $\overline{M}_{0,n}$ may be embedded into the product of projective spaces $\mathbb{P}^1\times \mathbb{P}^2\times \cdots \times \mathbb{P}^{n-3}$, using a combination of the Kapranov map $|ψ_n|:\overline{M}_{0,n}\to \mathbb{P}^{n-3}$ and the forgetful maps $π_i:\overline{M}_{0,i}\to \overline{M}_{0,i-1}$. We give an explicit combinatorial formula for the multidegree of this embedding in terms of certain parking functions of height $n-3$. We use this combinatorial interpretation to show that the total degree of the embedding (thought of as the projectivization of its cone in $\…
We introduce the class of transparent embeddings for a point-line geometry $Γ= ({\mathcal P},{\mathcal L})$ as the class of full projective embeddings $\varepsilon$ of $Γ$ such that the preimage of any projective line fully contained in $\varepsilon({\mathcal P})$ is a line of $Γ$. We will then investigate the transparency of Plücker embeddings of projective and polar grassmannians and spin embeddings of half-spin geometries and dual polar spaces of orthogonal type. As an application of our results on transparency, we will derive several Chow-like theorems for polar grassmannians and half-spin…
The aim is to give a geometric characterization of the finite generation of the Cox ring of anticanonical rational surfaces. This characterization is encoded in the finite generation of the effective monoid. Furthermore, we prove that in the case of a smooth projective rational surface having a negative multiple of its canonical divisor with only two linearly independent global sections (e.g., an elliptic rational surface), the finite generation is equivalent to the fact that there are only a finite number of smooth projective rational curves of self-intersection -1. The ground field is assume…
Let $T$ be a positive closed current of unit mass on the complex projective space $\mathbb P^n$. For certain values $α<1$, we prove geometric properties of the set of points in $\mathbb P^n$ where the Lelong number of $T$ exceeds $α$. We also consider the case of positive closed currents of bidimension (1,1) on multiprojective spaces.
We study hybrid models arising as homological projective duals (HPD) of certain projective embeddings $f:X\rightarrow\mathbb{P}(V)$ of Fano manifolds $X$. More precisely, the category of B-branes of such hybrid models corresponds to the HPD category of the embedding $f$. B-branes on these hybrid models can be seen as global matrix factorizations over some compact space $B$ or, equivalently, as the derived category of the sheaf of $\mathcal{A}$-modules on $B$, where $\mathcal{A}$ is an $A_{\infty}$ algebra. This latter interpretation corresponds to a noncommutative resolution of $B$. We compute…
In the first part of this paper, we show that the assertion "T_pH_k(X,Q)=G_pH_k(X,Q)" (which is called the Friedlander-Mazur conjecture) is a birationally invariant statement for smooth projective varieties X when p=dim(X)-2 and when p=1. We also establish the Friedlander-Mazur conjecture in certain dimensions. More precisely, for a smooth projective variety X, we show that the topological filtration T_pH_{2p+1}(X,Q) coincides with the geometric filtration G_pH_{2p+1}(X,Q) for all p. (Friedlander and Mazur had previously shown that T_pH_{2p}(X,Q})=G_pH_{2p}(X,Q)). As a corollary, we conclude t…
For arbitrary quantizable compact Kaehler manifolds, relations between the geometry given by the coherent states based on the manifold and the algebraic (projective) geometry realised via the coherent state mapping into projective space, are studied. Polar divisors, formulas relating the scalar products of coherent vectors on the manifold with the corresponding scalar products on projective space (Cauchy formulas), two-point, three-point and more generally cyclic m-point functions are discussed. The three-point function is related to the shape invariant of geodesic triangles in projective spac…
In arXiv:math/0405373 , Eisenbud, Huneke and Ulrich conjectured a result on the Castelnuovo-Mumford regularity of the embedding of a projective space $\mathbb{P}^{n-1}\hookrightarrow \mathbb{P}^{r-1}$ determined by generators of a linearly presented $\mathfrak{m}$-primary ideal. This result implies in particular that the image is scheme defined by equations of degree at most $n$. In this text we prove that the ideal of maximal minors of the Jacobian dual matrix associated to the input ideal defines the image as a scheme; it is generated in degree $n$. Showing that this ideal has a linear resol…
We study the projective behavior, mainly with respect to osculating spaces and secant varieties, of Lagrangian Grassmannians and Spinor varieties. We prove that these varieties have osculating dimension smaller than expected. Furthermore, we give numerical conditions ensuring the non secant defectivity of Lagrangian Grassmannians in their Plücker embedding and of Spinor varieties in both their Plücker and Spinor embeddings.
For certain complex projective manifolds (such as K3 surfaces and their higher dimensional analogues, the complex symplectic projective manifolds) the period map takes values in a locally symmetric variety of type IV. It is often an open embedding and in such cases it has been observed that the image is the complement of a locally symmetric (Heegner) divisor. We explain that phenomenon and get our hands on the complementary divisor in terms of geometric data.
The aim of this paper is to study geometric properties of non-degenerate smooth projective varieties of small degree from a birational point of view. First, using the positivity property of double point divisors and the adjunction mappings, we classify smooth projective varieties in $\mathbb P^r$ of degree $d \leq r+2$, and consequently, we show that such varieties are simply connected and rationally connected except in a few cases. This is a generalization of P. Ionescu's work. We also show the finite generation of Cox rings of smooth projective varieties in $\mathbb P^r$ of degree $d \leq r$…
Let $X$ be an arbitrary projective scheme over a field $k$. Let $A$ be the local ring at the vertex of the affine cone for some embedding $ι: X\hookrightarrow \mathbb{P}^n_k$. G. Lyubeznik asked (in \cite{l2}) whether the integers $λ_{i,j}(A)$ (defined in \cite{l1}), called the Lyubeznik numbers of $A$, depend only on $X$, but not on the embedding. In this paper, we make a big step toward a positive answer to this question by proving that in positive characteristic, for a fixed $X$, the Lyubezink numbers $λ_{i,j}(A)$ of the local ring $A$, can only achieve finitely many possible values under a…
Let $X$ be a projective scheme over a field $k$ and let $A$ be the local ring at the vertex of the affine cone of $X$ under some embedding $X\hookrightarrow\mathbb{P}^n_k$. We prove that, when $\ch(k)>0$, the Lyubeznik numbers $λ_{i,j}(A)$ are intrinsic numerical invariants of $X$, i.e., $λ_{i,j}(A)$ depend only on $X$, but not on the embedding.
Let $X \subset \mathbb{P}^{N-1}$ be a smooth projective variety. To each $g \in SL (N , \mathbb{C})$ which induces the embedding $g \cdot X \subset \mathbb{P}^{N-1}$ given by the ambient linear action we can associate a matrix $\barμ_X (g)$ called the centre of mass, which depends nonlinearly on $g$. With respect to the probability measure on $SL (N , \mathbb{C})$ induced by the Haar measure and the Gaussian unitary ensemble, we prove that the expectation of the centre of mass is a constant multiple of the identity matrix for any smooth projective variety.
Consider a projective manifold X and suppose that some wedge power of the cotangent bundle contains a subsheaf whose determinant bundle has maximal Kodaira dimension. Then we prove that X is of general type. More generally we compute the Kodaira dimension if the determinant bundle has sufficiently large Kodaira dimension. This is based on the study of the determinant bundle of a quotient of the cotangent bundle of a non-uniruled manifold: this bundle is always pseudo-effective. We apply this to study the universal cover of a projective manifold. Finally we prove the following: if the canonical…
