Anticanonical Divisor Research Articles

On $ℚ$-Conic Bundles, III

A ℚ -conic bundle germ is a proper morphism from a threefold with only terminal singularities to the germ (Z\ni o) of a normal surface such that fibers are connected and the anti-canonical divisor is relatively ample. Building upon our previous paper [MP08a], we prove the existence of a Du Val anti-canonical member under the assumption that the central fiber is irreducible.

Gluing construction of compact complex surfaces with trivial canonical bundle

We obtain a new construction of compact complex surfaces with trivial canonical bundle. In our construction we glue together two compact complex surfaces with an anticanonical divisor under suitable conditions. Then we show that the resulting compact manifold admits a complex structure with trivial canonical bundle by solving an elliptic partial differential equation. We generalize this result to cases where we have other than two components to glue together. With this generalization, we construct examples of complex tori, Kodaira surfaces and K3 surfaces. Lastly we deal with the smoothing problem of a normal crossing complex surface X with at most double curves. We prove that we still have a family of smoothings of X in a weak sense even when X is not Kahlerian or H 1 ( X , O X ) ≠ 0 , in which cases the smoothability result of Friedman [Fr] is not applicable.

Three-Dimensional Toric Morphisms with Anti-Nef Canonical Divisors

In this article, we classify projective toric birational morphisms from Gorenstein toric 3-folds onto the 3-dimensional affine space with relatively ample anti-canonical divisors.

On $ℚ$-conic Bundles, II

A ℚ -conic bundle germ is a proper morphism from a threefold with only terminal singularities to the germ (Z\ni o) of a normal surface such that fibers are connected and the anti-canonical divisor is relatively ample. We obtain the complete classification of ℚ -conic bundle germs when the base surface germ is singular. This is a generalization of [MP08], which further assumed that the fiber over o is irreducible.

On $ℚ$-conic Bundles

A ℚ -conic bundle is a proper morphism from a threefold with only terminal singularities to a normal surface such that fibers are connected and the anti-canonical divisor is relatively ample. We study the structure of ℚ -conic bundles near their singular fibers. One corollary to our main results is that the base surface of every ℚ -conic bundle has only Du Val singularities of type A (a positive solution of a conjecture by Iskovskikh). We obtain the complete classification of ℚ -conic bundles under the additional assumption that the singular fiber is irreducible and the base surface is singular.

An integrable system of K3-Fano flags

Given a K3 surface S, we show that the relative intermediate Jacobian of the universal family of Fano 3-folds V containing S as an anticanonical divisor is a Lagrangian fibration.

On the Crepancy of the Gieseker-Uhlenbeck Morphism

The Gieseker-Uhlenbeck morphism from the moduli space of Gieseker semistable rank-2 sheaves over an algebraic surface to the Uhlenbeck compactification was constructed by Jun Li. We prove that if the anti-canonical divisor of the surface is effective and the first Chern class of the semistable sheaves is odd, then the Gieseker-Uhlenbeck morphism is crepant

On invariance and Ricci-flatness of Hermitian metrics on open manifolds

We discuss a technique to construct Ricci-flat hermitian metrics on complements of (some) anticanonical divisors of almost homogeneous complex manifolds and inquire into when this metric is complete and Kähler. This construction has a strong interplay with invariance groups of the same dimension as the manifold acting with an open orbit. Lie groups of this type we call divisorial. As an application we describe compact manifolds admitting a divisorially invariant Kähler metric on an open subset. Finally, we see a connection between the reducibility of the anticanonical divisor and the non-triviality of the Kähler cone on the complement.

On surfaces of class VII + 0 with numerically anticanonical divisor

We consider minimal compact complex surfaces S with Betti numbers b 1 = 1 and n = b 2 > 0. A theorem of Donaldson gives n exceptional line bundles. We prove that if in a deformation, these line bundles have sections, S is a degeneration of blown-up Hopf surfaces. Besides, if there exists an integer m ≥ 1 and a flat line bundle F such that H 0 ( S,-mK ⊗ F ) ≠ 0, then S contains a Global Spherical Shell. We apply this last result to complete classification of bihermitian surfaces.

Multiplication Maps of Complete Linear Systems on Projective Toric Surfaces

We obtain an algebro-geometric proof of the surjectivity of the multiplication map of complete linear systems of ample divisor and nef divisor on a nonsingular projective toric surface. As a consequence, we show that on a nonsingular toric Fano 3-fold, the multiplication map of complete linear systems of the anti-canonical divisor and a nef divisor is surjective.

On projective varieties with nef anticanonical divisors

On projective varieties with nef anticanonical divisors

Irreducibility of the $(-1)$-classes on smooth rational surfaces

We give a characterization for a (-1)-divisor D on a smooth rational surface X to be irreducible under the assumption that an anticanonical divisor -K X of X is nef. Here -K X is nef means K X .C < 0 for every effective divisor C on X, and a (-1)-divisor D is a divisor such that the two numerical conditions D 2 = -1 = D.K X hold. As an application we give explicit examples of blowing up the projective plane at nine points infinitely near such that the obtained surface has an infinite number of (-1)-curves. A (-1)-curve is a smooth rational curve of self-intersection -1.

On Pseudo-Effectivity of the Second Chern Classes for Terminal Threefolds

We give a reduction of the conjecture that for terminal projective threefolds whose anticanonical divisors are nef, the second Chern classes are pseudo-effective. On the other hand, some effective non-vanishing results are obtained as applications of the pseudo-effectivity of the second Chern classes.

Exceptional curves on smooth rational surfaces with $-K$ not nef and of self-intersection zero

A (-n)-curve is a smooth rational curve of self-intersection -n, where n is a positive integer. In 1998 Hirschowitz asked whether a smooth rational surface X defined over the field of complex numbers, having an anti-canonical divisor not nef and of self-intersection zero, has (-2)-curves. In this paper we prove that for such a surface X, the set of (-1)-curves on X is finite but non-empty, and that X may have no (-2)-curves. Related facts are also considered.

A Note on Anti-Pluricanonical Maps for 5-Folds

We prove that the anti-pluricanonical map $\Phi_{|-mK_{X}|}$ is birational when $m\geq 16$ for $5$-fold $X$ whose anticanonical divisor is nef and big.

On pseudo-effectivity of the second chern classes for smooth threefolds

We prove that for smooth projective threefolds whose anticanonical divisors are nef, the second Chern classes are pseudo-effective under a weak assumption. As an application, the pseudo-effectivity of the second Chern classes implies that Kawamata’s Effective Non-vanishing Conjecture holds for such threefolds.

Classification of normal quartic surfaces with irrational singularities

If a normal quartic surface admits a singular point that is not a rational double point, then the surface is determined by the triplet (M,D,E) consisting of the minimal desingularization M, the pullback D of a general hyperplane section, and a nonzero effective anti-canonical divisor E of M. Geometric constructions of all the possible triplets (M,D,E) are given.

Anticanonical divisors of a moduli space of parabolic vector bundles of half weight on ℙ &lt;sup&gt;1&lt;/sup&gt;

Anticanonical divisors of a moduli space of parabolic vector bundles of half weight on ℙ &lt;sup&gt;1&lt;/sup&gt;

STABLE RANK-2 BUNDLES ON CALABI–YAU MANIFOLDS

In this paper, we apply the technique of chamber structures of stability polarizations to construct the full moduli space of rank-2 stable sheaves with certain Chern classes on Calabi–Yau manifolds which are anti-canonical divisor of ℙ1×ℙn or a double cover of ℙ1×ℙn. These moduli spaces are isomorphic to projective spaces. As an application, we compute the holomorphic Casson invariants defined by Donaldson and Thomas.

Hilbert functions of irreducible arithmetically Gorenstein schemes

In this paper we compute the Hilbert functions of irreducible (or smooth) and reduced arithmetically Gorenstein schemes that are twisted anti-canonical divisors on arithmetically Cohen–Macaulay schemes. We also prove some folklore results characterizing the Hilbert functions of irreducible standard determinantal schemes, and we use them to produce a new class of functions that occur as Hilbert functions of irreducible (or smooth) and reduced arithmetically Gorenstein schemes in any codimension.