Very ample line bundles on regular surfaces obtained by projection

Let L be a very ample line bundle on a smooth complex projective surface. The relations between the higher order embedding properties of L, like k ‐ spannedness, k ‐ very ampleness, k ‐ jet ampleness, and the higher numerical connectedness properties of divisors D ε |L| are Investigated. Some applications to adjoint bundles as well as pluricanonical bundles are discussed.

Let ϵ be an ample vector bundle of rank r ≥ 2 on a smooth complex projective variety X of dimension n such that there exists a global section of ϵ whose zero locus Z is a smooth subvariety of dimension n-r ≥ 2 of X. Let H be an ample line bundle on X such that the restriction HZ of H to Z is very ample. Triplets (X, ϵ, H) with g(Z, HZ) = 3 are classified, where g(Z, HZ) is the sectional genus of (Z, HZ).

Bridgeland stability of line bundles on surfaces

The strongest notion is k-jet ampleness; it implies k-very ampleness (cf. [BeSo2, Proposition 2.2]) which of course implies k-spannedness. For k = 0 or k = 1 all the three notions are equivalent and correspond to global generation resp. very ampleness. In this note we give criteria for k-jet ampleness of line bundles on abelian varieties. A naive way to obtain such a criterion is as follows: According to [BeSo2, Corollary 2.1] a tensor product of k very ample line bundles is always k-jet ample. Now on an abelian variety, by the generalization of Lefschetz’ classical theorem [LB,

We prove two new results for Seshadri constants on surfaces of general type. Let X be a surface of general type. In the first part, inspired by Bauer and Szemberg (Manuscripta Math 126(2):167–175, 2008), we list the possible values for the multi-point Seshadri constant $$\varepsilon (K_X,x_1,x_2,\ldots ,x_r)$$ when it lies between 0 and 1/r, where $$K_X$$ is the canonical line bundle on X. In the second part, we assume X of the form $$C \times C$$ , where C is a general smooth curve of genus $$g \geqslant 2$$ . Given such X and an ample line bundle L on X with some conditions on it, we show that the global Seshadri constant of L is a rational number.

In this paper, we study pushforwards of log pluricanonical bundles on projective log canonical pairs $(Y,\Delta)$ over the complex numbers, partially answering a Fujita-type conjecture due to Popa and Schnell in the log canonical setting. We show two effective global generation results. First, when $Y$ surjects onto a projective variety, we show a quadratic bound for generic generation for twists by big and nef line bundles. Second, when $Y$ is fibered over a smooth projective variety, we show a linear bound for twists by ample line bundles. These results additionally give effective non-vanishing statements. We also prove an effective weak positivity statement for log pluricanonical bundles in this setting, which may be of independent interest. In each context we indicate over which loci positivity holds. Finally, using the description of such loci, we show an effective vanishing theorem for pushforwards of certain log-sheaves under smooth morphisms.

Let pi :mathcal {X}rightarrow M be a holomorphic fibration with compact fibers and L a relatively ample line bundle over mathcal {X}. We obtain the asymptotic of the curvature of L^2-metric and Qullien metric on the direct image bundle pi _*(L^kotimes K_{mathcal {X}/M}) up to the lower order terms than k^{n-1}, for large k. As an application we prove that the analytic torsion tau _k(bar{partial }) satisfies partial bar{partial }log (tau _k(bar{partial }))^2=o(k^{n-1}), where n is the dimension of fibers.

In this work, a minimally FIIQ divisor on a smooth curve in is defined by an effective divisor which fails to impose independent conditions on quadrics in and all of whose proper subdivisors impose independent conditions on quadrics. Using this notion, we explicitly describe very ample non-special line bundles which fail to be normally generated on a simple -fold covering in terms of line bundles on under some numerical constraints. Furthermore, in case we obtain a concrete classification of such line bundles on

In this article, we determine the existing condition of cylinders in smooth minimal geometrically rational surfaces over a perfect field. Furthermore, we show that for any birational map between smooth projective surfaces, one contains a cylinder if and only if so does the other.

This chapter contains the basic theory of positivity for line bundles and divisors on a projective algebraic variety.

We construct a smooth complex projective rational surface with infinitely many mutually non-isomorphic real forms. This gives the first definite answer to a long-standing open question if a smooth complex projective rational surface has only finitely many non-isomorphic real forms or not.

We study when the Picard group of smooth surfaces of degree $d\geq 5$ in $\mathbb{P}^{3}$ acquires extra classes. In particular we show that the so-called exceptional components of the Noether–Lefschetz locus are not Zariski dense. This answers a 1991 question of C. Voisin. We also obtain similar results for the Noether–Lefschetz locus for suitable $(Y,L)$, where $Y$ is a smooth projective three-fold and $L$ a very ample line bundle. Both results are applications of the Zilber–Pink viewpoint recently developed by the authors for arbitrary (polarized, integral) variations of Hodge structures.

The Bordiga surface as critical locus for 3-view reconstructions

Let $H$ and $H'$ be two ample line bundles over a nonsingular projective surface $X$, and $M(H)$ (resp. $M(H')$) the coarse moduli scheme of $H$-semistable (resp. $H'$-semistable) sheaves of fixed type $(r=2,c_1,c_2)$. In a moduli-theoretic way that comes from elementary transforms, we connect $M(H)$ and $M(H')$ by a sequence of blowing-ups when walls separating $H$ and $H'$ are not necessarily good. As an application, we also consider the polarization change problem of Donaldson polynomials.

Let\(\mathcal{G}\) be the set of surfaces,S, polarized by a k-very ample line bundle,L, with genus≤3k+1. All the elements (S, L) of\(\mathcal{G}\) are listed. The classification of surfaces polarized by ak-very ample line bundle of degree ≤4k+4 is completed by proving that this class of surfaces is a subset of\(\mathcal{G}\).