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 resolution would imply that the conjecture in arXiv:math/0405373 holds. Furthermore, if this ideal of minors coincides with the one of the image in degree $n$ - what we hope to be true - the linearity of the resolution of this ideal of maximal minors is equivalent to the conjecture in arXiv:math/0405373.
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.
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.
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…
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.
For a smooth curve of genus $g$ embedded by a line bundle of degree at least $2g+3$ we show that the ideal sheaf of the secant variety is 5-regular. This bound is sharp with respect to both the degree of the embedding and the bound on the regularity. Further, we show that the secant variety is projectively normal for the generic embedding of degree at least $2g+3$. We also give a conjectural description of the resolutions of the ideals of higher secant varieties.
In this paper, we determine completely the Lyubeznik numbers $λ_{i,j}(A)$ of the local ring $A$ at the vertex of the affine cone over a nonsingular projective variety $V$, where $V$ is defined over a field of characteristic zero, in terms of the dimensions of the algebraic de Rham cohomology spaces of $V$. In particular, we prove that these numbers are intrinsic numerical invariants of $V$, even though a priori their definition depends on an embedding into projective space. This provides supporting evidence for a positive answer to the question of embedding-independence for arbitrary varieties…
Let X be a smooth projective variety and let K be the canonical divisor of X. In this paper, we study embeddings of X given by adjoint line bundles of the form K+L, where L is an ample line bundle. When X is a regular surface (i.e. H^1(X, O_X) = 0), we obtain a numerical criterion for K+L to have property Np. When X is a regular variety of arbitrary dimension, under a mild condition, we give an explicit calculation for the regularity of the ideal sheaves of such embeddings.
We study the number of (set-theoretically) defining equations of Segre products of projective spaces times certain projective hypersurfaces, extending results by Singh and Walther. Meanwhile, we prove some results about the cohomological dimension of certain schemes. In particular, we solve a conjecture of Lyubeznik about an inequality involving the cohomological dimension and the etale cohomological dimension of a scheme, in the characteristic-zero-case and under a smoothness assumption. Furthermore, we show that a relationship between depth and cohomological dimension discovered by Peskine a…
We survey recent results on open embeddings of the affine space $\mathbb{C}^n$ into a complete algebraic variety $X$ such that the action of the vector group $\mathbb{G}_a^n$ on $\mathbb{C}^n$ by translations extends to an action of $\mathbb{G}_a^n$ on $X$. We begin with Hassett-Tschinkel correspondence describing equivariant embeddings of $\mathbb{C}^n$ into projective spaces and give its generalization for embeddings into projective hypersurfaces. Further sections deal with embeddings into flag varieties and their degenerations, complete toric varieties, and Fano varieties of certain types.
The aim of this paper is twofold: First we classify all abstract light dual multinets of order $6$ which have a unique line of length at least two. Then we classify the weak projective embeddings of these objects in projective planes over fields of characteristic zero. For the latter we present a computational algebraic method for the study of weak projective embeddings of finite point-line incidence structures.
We prove that every projective embedding of a connected scheme determined by the complete linear series of a sufficiently ample line bundle is defined by the 2-minors of a 1-generic matrix of linear forms. Extending the work of Eisenbud-Koh-Stillman for integral curves, we also provide effective descriptions for such determinantally presented ample line bundles on products of projective spaces, Gorenstein toric varieties, and smooth n-folds.
Let P^n be the projective space of dimension n on an algebraically closed field of characteristic 0 and F be the image of the Segre embedding of P^1xP^1xP^1 inside P^7. In the present paper we deal with the moduli spaces of locally free sheaves E on F of rank 2 with h^i(F,E(t))=0 for i=1,2 and each integer t.
We study matrix factorizations of locally free coherent sheaves on a scheme. For a scheme that is projective over an affine scheme, we show that homomorphisms in the homotopy category of matrix factorizations may be computed as the hypercohomology of a certain mapping complex. Using this explicit description, we give another proof of Orlov's theorem that there is a fully faithful embedding of the homotopy category of matrix factorizations into the singularity category of the corresponding zero subscheme. We also give a complete description of the image of this functor.
This paper builds on work of Hochster and Yao that provides nice embeddings for finitely generated modules of finite G-dimension, finite projective dimension, or locally finite injective dimension. We extend these results by providing similar embeddings in the relative setting, that is, for certain modules of finite G_C-dimension, finite P_C-projective dimension, locally finite GI_C-injective dimension, or locally finite I_C-injective dimension where C is a semidualizing module. Along the way, we extend some results for modules of finite homological dimension to modules of locally finite homol…
Totally skew embeddings are introduced by Ghomi and Tabachnikov. They are naturally related to classical problems in topology, such as the generalized vector field problem and the immersion problem for real projective spaces. In recent paper Topological obstructions to totally skew embeddings, {totaly skew} embeddings are studied by using the Stiefel-Whitney classes In the same paper it is conjectured that for every $n$-dimensional, compact smooth manifold $M^n$ $(n>1)$, $$N(M^n)\leq 4n-2α(n)+1,$$ where $N(M^n)$ is defined as the smallest dimension $N$ such that there exists a {\em totally ske…
