Adrian Langer
We prove Reider type criterions for k-jet spannedness and k-jet ampleness of adjoint bundles for surfaces with at most rational singularities. Moreover, we prove that on smooth surfaces [n(n+4)/4]-very ampleness implies n-jet ampleness.
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We study k-jet ampleness of line bundles on hyperelliptic surfaces using vanishing theorems. Our main result states that on a hyperelliptic surface of an arbitrary type a line bundle of type (m,m) with m\geq k+2 is k-jet ample.
A celebrated conjecture of Kobayashi and Lang says that the canonical line bundle $K_X$ of a Kobayashi hyperbolic compact complex manifold $X$ is ample. In this note we prove that $K_X$ is ample if $X$ is projective and satisfies a stronger condition of nondegenerate negative total $k$-jet curvature. We use positivity of direct image sheaves and decomposition of jets in order to produce pluridifferentials on $X$.
We study k-very ampleness of line bundles on blow-ups of hyperelliptic surfaces at r very general points. We obtain a numerical condition on the number of points for which a line bundle on the blow-up of a hyperelliptic surface at these r points gives an embedding of order k.
In this note we consider the blowing-up X of P^3 along r general points of the anticanonical divisor of a smooth quadric in P^3. Given a complete linear system |L| = |dH - m_1 E_1 -...- m_r E_r| on X, with H the pull-back of a plane in P^3 and E_i the exceptional divisor corresponding to P_i, we give necessary and sufficient conditions for the very ampleness (resp. base point freeness and non-speciality) of L. As a corollary we obtain a sufficient condition for the very ampleness of such a complete linear system on the blowing-up of P^3 along r general points.
Jet ampleness of line bundles generalizes very ampleness by requiring the existence of enough global sections to separate not just points and tangent vectors, but also their higher order analogues called jets. We give sharp bounds guaranteeing that a line bundle on a projective toric variety is $k$-jet ample in terms of its intersection numbers with the invariant curves, in terms of the lattice lengths of the edges of its polytope, in terms of the higher concavity of its piecewise linear function and in terms of its Seshadri constant. For example, the tensor power $k+n-2$ of an ample line bund…
In this short note, we consider the conjecture that the log canonical divisor (resp. the anti-log canonical divisor) $K_X + Δ$ (resp. $-(K_X + Δ)$) on a pair $(X, Δ)$ consisting of a complex projective manifold $X$ and a reduced simply normal crossing divisor $Δ$ on $X$ is ample if it is numerically positive. More precisely, we prove the conjecture for $K_X + Δ$ with $Δ\neq 0$ in dimension 4 and for $-(K_X + Δ)$ with $Δ\neq 0$ in dimension 3 or 4.
In this note, we prove effective semi-ampleness conjecture due to Prokhorov and Shokurov for a special case, more concretely, for Q-Gorenstein klt-trivial fibrations over smooth projective curves whose fibers are all klt log Calabi-Yau pairs of Fano type.
Obstructions to the ampleness of invariant jet differentials on complete intersections
该研究证明在四维射影空间中光滑超曲面上，特定阶和权重的不变射流微分丛不丰沛，否证了相关猜想，并推广到完全交和更高阶情形。
Higher order embeddings via the basepoint-freeness threshold
本文建立极化阿贝尔簇的无基点阈值与 k-射流极丰性的联系，并据此给出包括 Kummer 簇 k-极丰性判据在内的应用。
In this note, we apply the semi-ampleness criterion in Lemma 3.1 to prove many classical results in the study of abundance conjecture. As a corollary, we prove abundance for large Kodaira dimension depending only on [BCHM10].
Moving Seshadri constants and effective Fujita-type conjectures
本文提出并探讨了新的 Fujita 猜想变体，涉及线性系统形式 |K_X + aK_X + bL| 的无基点性、非常量性和喷射放大性，并扩展了移动塞沙迪常数理论至任意域上的完备变种。
In this note we study properties of partially ample line bundles on simplicial projective toric varieties. We prove that the cone of q-ample line bundles is a union of rational polyhedral cones, and calculate these cones in examples. We prove a restriction theorem for big q-ample line bundles, and deduce that q-ampleness of the anticanonical bundle is not invariant under flips. Finally we prove a Kodaira-type vanishing theorem for q-ample line bundles.
In this article we briefly discuss the finite generation of fiber rings of invariant k-jets of holomorphic curves in a complex projective manifold, using differential Galois theory.
We formulate and prove the existence of an asymptotic duality along the fibers of the Green-Griffiths jet bundles over projective manifolds. The existence of global sections for these bundles and also for their dual sheaves has been established by the author and J. P. Demailly in a previous article, [3].
This note is intended to provide a general reference for jet spaces and jet differentials, valid in maximal generality (at the level of EGA). The approach is rather concrete, using Hasse-Schmidt (divided) higher differentials. Discussion of projectivized jet spaces (as in Green and Griffiths (1980)) is included.
We prove a product decomposition of the Zariski closure of the jet lifts of a holomorphic map f from C into a semi-abelian variety A, provided that f is of finite order. On the other hand, by giving an example of such a map f into a three dimensional abelian variety we show that this product decomposition does not hold in general; there was a gap in the proofs of some articles which claimed that such a product decomposition is always true.
M. Mustata used jet schemes to compute the multiplier ideals of reduced hyperplane arrangements. We give an alternate proof using a log resolution, which is simpler and allows us to consider non-reduced arrangements. By applying the idea of wonderful models introduced by De Concini--Procesi, we also simplify the result.
In this paper we give a complete description of the irreducible components of the jet schemes (with origin in the singular locus) of a two-dimensional quasi-ordinary hypersurface singularity. We associate with these components and with their codimensions and embedding dimensions, a weighted graph. We prove that the data of this weighted graph is equivalent to the data of the topological type of the singularity. We also determine a component of the jet schemes (or equivalently, a divisor on $\mathbb{A}^3$), that computes the log canonical threshold of the singularity embedded in $\mathbb{A}^3$.…
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