Codimension bounds for the Noether–Lefschetz components for toric varieties

For a quasi-smooth hypersurface X in a projective simplicial toric variety $$\mathbb {P}_{\Sigma }$$ P Σ , the morphism $$i^*:H^p(\mathbb {P}_{\Sigma })\rightarrow H^p(X)$$ i ∗ : H p ( P Σ ) → H p ( X ) induced by the inclusion is injective for $$p=\dim X$$ p = dim X and an isomorphism for $$p<\dim X-1$$ p < dim X - 1 . This allows one to define the Noether–Lefschetz locus $$\mathrm{NL}_{\beta }$$ NL β as the locus of quasi-smooth hypersurfaces of degree $$\beta $$ β such that $$i^*$$ i ∗ acting on the middle algebraic cohomology is not an isomorphism. We prove that, under some assumptions, if $$\dim \mathbb {P}_{\Sigma }=2k+1$$ dim P Σ = 2 k + 1 and $$k\beta -\beta _0=n\eta $$ k β - β 0 = n η , $$n\in \mathbb {N}$$ n ∈ N , where $$\eta $$ η is the class of a 0-regular ample divisor, and $$\beta _0$$ β 0 is the anticanonical class, every irreducible component V of the Noether–Lefschetz locus quasi-smooth hypersurfaces of degree $$\beta $$ β satisfies the bounds $$n+1\leqslant \mathrm{codim}\,Z \leqslant h^{k-1,\,k+1}(X)$$ n + 1 ⩽ codim Z ⩽ h k - 1 , k + 1 ( X ) .

Affine cones over cubic surfaces are flexible in codimension one

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.

The Noether–Lefschetz locus of surfaces in toric threefolds

The Noether–Lefschetz theorem asserts that any curve in a very general surface [Formula: see text] in [Formula: see text] of degree [Formula: see text] is a restriction of a surface in the ambient space, that is, the Picard number of [Formula: see text] is [Formula: see text]. We proved previously that under some conditions, which replace the condition [Formula: see text], a very general surface in a simplicial toric threefold [Formula: see text] (with orbifold singularities) has the same Picard number as [Formula: see text]. Here we define the Noether–Lefschetz loci of quasi-smooth surfaces in [Formula: see text] in a linear system of a Cartier ample divisor with respect to a [Formula: see text]-regular, respectively 0-regular, ample Cartier divisor, and give bounds on their codimensions. We also study the components of the Noether–Lefschetz loci which contain a line, defined as a rational curve which is minimal in a suitable sense.

CHARACTERIZATION OF PROJECTIVE SPACES AND -BUNDLES AS AMPLE DIVISORS

Let $X$ be a projective manifold of dimension $n$. Suppose that $T_{X}$ contains an ample subsheaf. We show that $X$ is isomorphic to $\mathbb{P}^{n}$. As an application, we derive the classification of projective manifolds containing a $\mathbb{P}^{r}$-bundle as an ample divisor by the recent work of Litt.

Semiample perturbations for log canonical varieties over an F-finite field containing an infinite perfect field

Let [Formula: see text] be an [Formula: see text]-finite field containing an infinite perfect field of positive characteristic. Let [Formula: see text] be a projective log canonical pair over [Formula: see text]. In this note, we show that, for a semi-ample divisor [Formula: see text] on [Formula: see text], there exists an effective [Formula: see text]-divisor [Formula: see text] such that [Formula: see text] is log canonical if there exists a log resolution of [Formula: see text].

CHARACTERIZATION OF

In this paper a new geometric characterization of the $n$th symmetric product of a curve is given. Specifically, we assume that there exists a chain of smooth subvarieties $V_{i}$ of dimension $i$, such that $V_{i}$ is an ample divisor in $V_{i+1}$ and its intersection product with $V_{1}$ is one; that the Albanese dimension of $V_{2}$ is $2$ and the genus of $V_{1}$ is equal to the irregularity of the variety. We prove that in this case the variety is isomorphic to the symmetric product of a curve.

Counterexamples of Lefschetz hyperplane type results for movable cones

The main theorem of the paper provides a way to produce examples such that the movable cone of an ample divisor does not coincide with the movable cone of its ambient variety.

Chern Classes of Splayed Intersections

We generalize the Chern class relation for the transversal intersection of two nonsingular varieties to a relation for possibly singular varieties, under a splayedness assumption. We show that the relation for the Chern–Schwartz–MacPherson classes holds for two splayed hypersurfaces in a nonsingular variety, and under a strong splayedness assumption for more general subschemes. Moreover, the relation is shown to hold for the Chern–Fulton classes of any two splayed subschemes. The main tool is a formula for Segre classes of splayed subschemes. We also discuss the Chern class relation under the assumption that one of the varieties is a general very ample divisor.

Every Curve of Genus not Greater Than Eight Lies on a K3 Surface

Let C be a smooth irreducible complete curve of genus g ≥ 2 over an algebraically closed field of characteristic 0. An ample K3 extension of C is a K3 surface with at worst rational double points which contains C in the smooth locus as an ample divisor.In this paper, we prove that all smooth curve of genera. 2 ≤ g ≤ 8 have ample K3 extensions. We use Bertini type lemmas and double coverings to construct ample K3 extensions.

Some Results on Surfaces of General Type

In this article we prove some new results on projective normality, normal presentation and higher syzygies for surfaces of general type, not necessarily smooth, embedded by adjoint linear series. Some of the corollaries of more general results include: results on property Np associated to KS ⊗B⊗n where B is base-point free and ample divisor with B⊗K* nef, results for pluricanonical linear systems and results giving effective bounds for adjoint linear series associated to ample bundles. Examples in the last section show that the results are optimal.