Singular Projective Varieties Research Articles

Reconstruction of Projective Curves from the Derived Category

We prove that integral projective curves with Cohen–Macaulay singularities over an algebraically closed field of characteristic zero are determined by their triangulated category of perfect complexes. This partially extends a theorem of Bondal and Orlov to the case of singular projective varieties and in particular shows that all integral complex projective curves are determined by their category of perfect complexes.

Curves contracted by the Gauss map

Given a singular projective variety in some projective space, we characterize the smooth curves contracted by the Gauss map in terms of normal bundles. As a consequence, we show that if the variety is not linear, then a contracted line always has local obstruction for the embedded deformation and each component of the Hilbert scheme where the line lies is non-reduced everywhere.

Zero-cycle groups on algebraic varieties

We compare various groups of 0-cycles on quasi-projective varieties over a field. As applications, we show that for certain singular projective varieties, the Levine-Weibel Chow group of 0-cycles coincides with the corresponding Friedlander-Voevodsky motivic cohomology. We also show that over an algebraically closed field of positive characteristic, the Chow group of 0-cycles with modulus on a smooth projective variety with respect to a reduced divisor coincides with the Suslin homology of the complement of the divisor. We prove several generalizations of the finiteness theorem of Saito and Sato for the Chow group of 0-cycles over p-adic fields. We also use these results to deduce a torsion theorem for Suslin homology which extends a result of Bloch to open varieties.

Euclidean Distance Degree of Projective Varieties

Abstract We give a positive answer to a conjecture of Aluffi–Harris on the computation of the Euclidean distance degree of a possibly singular projective variety in terms of the local Euler obstruction function.

Equations and Syzygies of K3 Carpets and Unions of Scrolls

We describe the equations and Grobner bases of some degenerate K3 surfaces associated to rational normal scrolls. These K3 surfaces are members of a class of interesting singular projective varieties we call correspondence scrolls. The ideals of these surfaces are nested in a simple way that allows us to analyze them inductively. We describe explicit Grobner bases and syzygies for these objects over the integers and this lets us treat them in all characteristics simultaneously.

Algebraic integrability of foliations with numerically trivial canonical bundle

Given a reflexive sheaf on a mildly singular projective variety, we prove a flatness criterion under certain stability conditions. This implies the algebraicity of leaves for sufficiently stable foliations with numerically trivial canonical bundle such that the second Chern class does not vanish. Combined with the recent works of Druel and Greb–Guenancia–Kebekus this establishes the Beauville–Bogomolov decomposition for minimal models with trivial canonical class.

La conjecture de Manin pour une famille de variétés en dimension supérieure

AbstractInspired by a method of La Bretèche relying on some unique factorisation, we generalise work of Blomer, Brüdern and Salberger to prove Manin's conjecture in its strong form conjectured by Peyre for some infinite family of varieties of higher dimension. The varieties under consideration in this paper correspond to the singular projective varieties defined by the following equation$$ x_1 y_2y_3\cdots y_n+x_2y_1y_3 \cdots y_n+ \cdots+x_n y_1 y_2 \cdots y_{n-1}=0 $$in ℙℚ2n−1for alln⩾ 3. This paper comes with an Appendix by Per Salberger constructing a crepant resolution of the above varieties.

Twisted cubics on singular cubic fourfolds—On Starr’s fibration

We show that the Hilbert scheme compactification of the total space of Starr’s fibration on the space of twisted cubics on a cubic hypersurface in $${\mathbb P}^5$$ not containing a plane admits a contraction to a singular projective symplectic variety of dimension eight which has a crepant resolution deformation equivalent to the symplectic eightfold constructed from twisted cubics on a smooth cubic fourfold. This yields another proof that the symplectic eightfold and the Hilbert scheme of four points on a K3 surface are deformation equivalent. As a byproduct we obtain similar results for the variety of lines on a singular cubic fourfold.

On the algebraic stringy Euler number

We are interested in stringy invariants of singular projective algebraic varieties satisfying a strict monotonicity with respect to elementary birational modifications in the Mori program. We conjecture that the algebraic stringy Euler number is one of such invariants. In the present paper, we prove this conjecture for varieties having an action of a connected algebraic group G and admitting equivariant desingularizations with only finitely many G-orbits. In particular, we prove our conjecture for arbitrary projective spherical varieties.

Extremal transition and quantum cohomology: Examples of toric degeneration

When a singular projective variety Xsing admits a projective crepant resolution Xres and a smoothing Xsm, we say that Xres and Xsm are related by extremal transition. In this article, we study a relationship between the quantum cohomology of Xres and Xsm in some examples. For 3-dimensional conifold transition, a result of Li and Ruan implies that the quantum cohomology of a smoothing Xsm is isomorphic to a certain subquotient of the quantum cohomology of a resolution Xres with the quantum variables of exceptional curves specialized to one. We observe that similar phenomena happen for toric degenerations of Fl(1,2,3), Gr(2,4), and Gr(2,5) by explicit computations.

Algebraic Versus Homological Equivalence for Singular Varieties

Let Y ⊆ ℙN be a possibly singular projective variety, defined over the field of complex numbers. Let X be the intersection of Y with h general hypersurfaces of sufficiently large degrees. Let d > 0 be an integer, and assume that dimY = n + h and dimYsing ≤ min {d + h − 1, n − 1}. Let Z be an algebraic cycle on Y of dimension d + h, whose homology class in H2(d+h)(Y; ℚ) is nonzero. In the present article, we prove that the restriction of Z to X is not algebraically equivalent to zero. This is a generalization to the singular case of a result due to Nori in the case Y is smooth. As an application we provide explicit examples of singular varieties for which homological equivalence is different from the algebraic one.

Existence of a singular projective variety with an arbitrary set of characteristic numbers

It is known that Chern characteristic numbers of compact complex manifolds cannot have arbitrary values. They satisfy certain divisability conditions. W. Ebeling and S. M. Gusein-Zade gave a definition of Chern characteristic numbers of singular compact complex analytic varieties. We prove that there exists a singular projective variety with an arbitrary set of characteristic numbers.

A geometric approach to problems in birational geometry

A classical set of birational invariants of a variety are its spaces of pluricanonical forms and some of their canonically defined subspaces. Each of these vector spaces admits a typical metric structure which is also birationally invariant. These vector spaces so metrized will be referred to as the pseudonormed spaces of the original varieties. A fundamental question is the following: Given two mildly singular projective varieties with some of the first variety's pseudonormed spaces being isometric to the corresponding ones of the second variety's, can one construct a birational map between them that induces these isometries? In this work, a positive answer to this question is given for varieties of general type. This can be thought of as a theorem of Torelli type for birational equivalence.

Roitman's theorem for singular projective varieties in arbitrary characteristic

AbstractIn this paper, we prove Roitman's theorem regarding torsion 0-cycles for singular projective varieties over algebraically closed fields of arbitrary characteristic, for torsion which is of exponent prime to the characteristic. This generalizes earlier results for complex projective varieties. Our proof even in that case is different from the earlier ones.

Generalized albanese and its dual

A generalization of the Albanese variety to the case of a singular projective variety X over an algebraically closed field k is given in [ESV], where H. Esnault, V. Srinivas and E. Viehweg constructed a universal regular quotient of the Chow group CH0(X)deg 0 of 0-cycles of degree 0 modulo rational equivalence. This is a smooth connected commutative algebraic group, universal for rational maps from X to smooth commutative algebraic groups which factor through a homomorphism of groups CH0(X)deg 0 ��! G(k). Suppose now that in addition k is of characteristic 0. Interpreting this algebraic group as a generalized 1-motive in the sense of Laumon [L], we may ask for the dual 1-motive. The intention of these notes was to describe the functor which is represented by the dual 1-motive. This forms the main result of this work. The notion of dual 1-motive allows to treat the problem in a more general way: We consider certain categories of rational maps from a projective variety to commutative algebraic groups (the category of rational maps factoring through CH0(X)deg 0 is a special case). A necessary and sufficient condition for the existence of an object of such a category satisfying the universal mapping property is given, as well as a construction of these universal objects via their dual 1-motives. In particular, this provides an independent proof of the existence and an explicit construction of the universal regular quotient for algebraically closed base field of characteristic 0.

NONEXISTENCE OF A CREPANT RESOLUTION OF SOME MODULI SPACES OF SHEAVES ON A K3 SURFACE

Let $M_c=M(2,0,c)$ be the moduli space of O(1)-semistable rank 2 torsion-free sheaves with Chern classes $c_1=0$ and $c_2=c$ on a K3 surface $X$ where O(1) is a generic ample line bundle on $X$. When $c=2n\geq4$ is even, $M_c$ is a singular projective variety equipped with a symplectic structure on the smooth locus. In this paper, we show that there is no crepant resolution of $M_{2n}$ for $n\geq 3$. This implies that there is no symplectic desingularization.

On the Gauss maps of singular projective varieties

AbstractHere we study the dimension δ(m, X) of the general fibers of the m-Gaussian map of a singular n-dimensional variety X ⊂ Pn. We show that for all integers a, b, c, d with n ≦ a < b ≦ c < d ≦ N − 1 and a + d = b + c we have δ (a, X) + δ(d, X) > δ(b, X) + δ(c, X). If δ(X, N − 1) is very large we give some classification results which extend to the singular case some results of Ein.

Roitman‘s Theorem for Singular Projective Varieties

We construct an Abel–Jacobi mapping on the Chow group of 0-cycles of degree 0, and prove a Roitman theorem, for projective varieties over C with arbitrary singularities. Along the way, we obtain a new version of the Lefschetz Hyperplane theorem for singular varieties.

The zeta function of a simplicial complex

Given a simplicial complex δ on vertices {1, …,n} and a fieldF we consider the subvariety of projective (n−1)-space overF consisting of points whose homogeneous coordinates have support in δ. We give a simple rational expression for the zeta function of this singular projective variety overF q and show a close connection with the Betti numbers of the corresponding variety over ℂ. This connection is particularly simple in the case when Δ is Cohen-Macaulay.

Moving algebraic cycles of bounded degree

The Chow Moving Lemma is a theorem which asserts that a given algebraic s-cycle on a smooth algebraic variety X can be moved within its rational equivalence class to intersect properly a given r-cycle on X provided that r + s ≥ dim(X) (cf. [Chow], [S2]). In the past few years, there has been considerable interest in studying spaces of algebraic cycles rather than simply cycles modulo an equivalence relation. With this in mind, it is natural to ask whether one can move a given “bounded family” of s-cycles on the smooth variety X to intersect properly a given “bounded family” of r-cycles. The main point of this paper is to formulate and prove just such a result. In Theorem 3.1, we demonstrate that for any integer e and any smooth projective variety X, one can simultaniously and algebraically “move” all effective s-cycles of degree ≤ e on X so that each such cycle meets every effective r-cycle of degree ≤ e on X in proper dimension. The primary motivation for this Moving Lemma for Cycles of Bounded Degree was the possibility of a duality theorem between cohomology and homology theories defined in terms of homotopy groups of cycle spaces. Using Theorem 3.1, we have proved such a duality theorem for complex quasi-projective varieties in [F-L2]. We prove our Moving Lemma for varieties over an arbitary infinite field, permitting a proof in [F-V] of a duality theorem for “motivic cohomology and homology”. The reader will find that our Moving Lemma has numerous good properties. First of all, the move is given as an algebraic move (parametrized by a punctured projective line) on Chow varietes. Although this move is “good” only for s-cycles of bounded degree, it is defined on all effective s-cycles. Moreover, the move starts at “time 0” by expressing an effective s-cycle Z as a difference of effective s-cycles both of which have intersection properties no worse than Z. Finally, our Moving Lemma is applicable to smooth quasiprojective varieties, for it is stated for a possibly singular projective variety X resulting in a conclusion of proper intersection off the singular locus of X. The classical motivation for the moving lemma was to define an intersection product on algebraic cycles modulo rational equivalence, thereby establishing the Chow ring A∗(X). Some of the classical literature overlooked the question of whether or not intersection of cycles defined via a moving lemma is independent of the move (e.g., [Chow], [R], [S2]; on the other hand, cf. [Chev], [S3])). One direct consequence of our Moving Lemma is a proof for smooth quasi-projective varieties that the intersection product is indeed well defined independent of the choice of move (Theorem 3.4). Of course, the intersection product now has an intrinsic formulation for all smooth algebraic varieties due to Fulton