A bound on the irregularity of abelian scrolls in projective space

Classification of Elliptic Line Scrolls

The classification of projective elliptic line scrolls with the description of their singular loci is given. In particular we recover Atiyah Theorem by using classical methods.

关于加权涡卷中 K3 纤维化 Calabi-Yau 三流形

On K3 fibred Calabi-Yau threefolds in weighted scrolls

本文分类了嵌入加权涡卷的低度加权 K3 曲面纤维化温和奇异 Calabi-Yau 三流形，推广 Mullet 结果，并研究射影退化与嵌入问题。

A Castelnuovo-Mumford regularity bound for scrolls

Let $X \subseteq \mathbb{P}^r$ be a scroll of codimension $e$ and degree $d$ over a smooth projective curve of genus $g$. The purpose of this paper is to prove a linear Castelnuovo-Mumford regularity bound that reg$(X) \leq d-e+1+g(e-1)$. This bound works over an algebraically closed field of arbitrary characteristic.

Quot schemes, Segre invariants, and inflectional loci of scrolls over curves

Let $E$ be a vector bundle over a smooth curve $C$, and $S = \mathbb{P} E$ the associated projective bundle. We describe the inflectional loci of certain projective models $ψ\colon S \dashrightarrow \mathbb{P}^n$ in terms of Quot schemes of $E$. This gives a geometric characterisation of the Segre invariant $s_1 (E)$, which leads to new geometric criteria for semistability and cohomological stability of bundles over $C$. We also use these ideas to show that for general enough $S$ and $ψ$, the inflectional loci are all of the expected dimension. An auxiliary result, valid for a general subvarie…

Hilbert scheme of some threefold scrolls over the Hirzebruch surface F_1

Hilbert schemes of suitable smooth, projective manifolds of low degree which are 3-fold scrolls over the Hirzebruch surface F_1 are studied. An irreducible component of the Hilbert scheme parametrizing such varieties is shown to be generically smooth of the expected dimension and the general point of such a component is described.

A Series of Smooth Irregular Varieties in Projective Space

One of the simplest examples of a smooth, non degenerate surface in P^4 is the quintic elliptic scroll. It can be constructed from an elliptic normal curve E by joining every point on E with the translation of this point by a non-zero 2-torsion point. The same construction can be applied when E is replaced by a (lineaerly normally embedded) abelian variety A. In this paper we ask the question when the resulting scroll Y is smooth. If A is an abelian surface embedded by a line bundle L of type (d_1,d_2) and r=d_1d_2, then we prove that for general A the scroll Y is smooth if r is at least 7 wit…

Hilbert schemes of some threefold scrolls over F_e

Hilbert schemes of suitable smooth, projective 3-fold scrolls over the Hirzebruch surface F_e, with e > 1, are studied. An irreducible component of the Hilbert scheme parametrizing such varieties is shown to be generically smooth of the expected dimension and the general point of such a component is described. This article generalizes the study of Hilbert schemes done in arXiv:1110.5464 for e=1.

Ulrich bundles on smooth projective varieties of minimal degree

We classify the Ulrich vector bundles of arbitrary rank on smooth projective varieties of minimal degree. In the process, we prove the stability of the sheaves of relative differentials on rational scrolls.

Moduli spaces of bundles and Hilbert schemes of scrolls over $ν$-gonal curves

The aim of this paper is two--fold. We first strongly improve our previous main result Theorem 3.1 in Arxiv 1702.00918v3 12Feb2018 ("Brill-Noether loci of rank two vector bundles on a general $ν$-gonal curve"), concerning classification of irreducible components of the Brill--Noether locus parametrizing rank 2 semistable vector bundles of suitable degrees $d$, with at least $d-2g+4$ independent global sections, on a general $ν$--gonal curve $C$ of genus $g$. We then uses this classification to study several properties of the Hilbert scheme of suitable surface scrolls in projective space, which…

Projective surfaces of maximal sectional regularity

We study projective surfaces $X \subset \mathbb{P}^r$ (with $r \geq 5$) of maximal sectional regularity and degree $d > r$, hence surfaces for which the Castelnuovo-Mumford regularity $\reg(C)$ of a general hyperplane section curve $C = X \cap \mathbb{P}^{r-1}$ takes the maximally possible value $d-r+3$. We show that each of these surfaces is either a cone over a curve $C \subset \mathbb{P}^{r-1}$ of maximal regularity or else a birational outer linear projection of a smooth rational surface scroll $\widetilde{X} \subset \mathbb{P}^{d+1}$. We prove that the Castelnuovo-Mumford regularity of th…

关于具有 Picard 数 2 的若干射影簇上的本原 Ulrich 丛

On Primitive Ulrich Bundles over a few projective varieties with Picard number two

本文在光滑射影簇中引入本原 Ulrich 丛的概念，并在次数 $6$ 的旗三维簇和有理法向滚动面情形下给出了其上同调刻画，同时提出了若干开放问题。

三重固体与卷积

Triple solids and scrolls

研究光滑射影簇在度3有限态射下的卷积结构，证明维度限制并分类m=1和m=2情形，发现多种情况简化为Segre积。

On The Projective Normality of Smooth Surfaces of degree nine

We investigate the projective normality of smooth, linearly normal surfaces of degree 9. All non projectively normal surfaces which are not scrolls over a curve are classified. Results on the projective normality of surface scrolls are also given. One of the reasons that brought us to look at this question is our desire to find examples for a long standing problem in adjunction theory. Andreatta followed by a generalization by Ein and Lazarsfeld posed the problem of classifying smooth n-dimensional varieties (X,L) polarized with a very ample line bundle L, such that the adjoint linear system |…

射影空间中非常一般超曲面的代数双曲性

Algebraic hyperbolicity of very general hypersurfaces in projective spaces

本文证明了射影空间中非常一般超曲面代数双曲性的最后未解情形，即四次三维簇，完成了分类，方法基于Coskun--Riedl构造及稳定映射模空间和对数几何。

Instanton sheaves on projective schemes

A $h$-instanton sheaf on a closed subscheme $X$ of some projective space endowed with an ample and globally generated line bundle $\mathcal{O}_X(h)$ is a coherent sheaf whose cohomology table has a certain prescribed shape. In this paper we deal with $h$-instanton sheaves relating them to Ulrich sheaves. Moreover, we study $h$-instanton sheaves on smooth curves and surfaces, cyclic $n$-folds, Fano $3$-folds and scrolls over arbitrary smooth curves. We also deal with a family of monads associated to $h$-instanton bundles on varieties satisfying some mild extra technical conditions.

The dimension of the Hilbert scheme of special threefolds

The Hilbert scheme of projective 3-folds of codimension 3 or more that are linear scrolls over the projective plane or over a smooth quadric surface or that are quadric or cubic fibrations over the projective line is studied. All known such threefolds of degree from 7 to 11 are shown to correspond to smooth points of an irreducible component of their Hilbert scheme, whose dimension is computed. A relationship with the locus of good determinantal subschemes is investigated

Surfaces of minimal degree of tame representation type and mutations of Cohen-Macaulay modules

We provide two examples of smooth projective surfaces of tame CM type, by showing that any parameter space of isomorphism classes of indecomposable ACM bundles with fixed rank and determinant on a rational quartic scroll in projective 5-space is either a single point or a projective line. For surfaces of minimal degree and wild CM type, we classify rigid Ulrich bundles as Fibonacci extensions. For the rational normal scrolls S(2,3) and S(3,3), a complete classification of rigid ACM bundles is given in terms of the action of the braid group in three strands.

半阿贝尔簇的有效刻画

Effective characterization of semi-abelian varieties

研究任意维复半阿贝尔簇的有效刻画，利用对数不规则性和前两个对数多重亏格，在特定条件簇中得出相关结果，是任意维拟射影簇该方向首个成果。

Characterization of Abelian Varieties

We prove that any smooth complex projective variety $X$ with plurigenera $P_1(X)=P_2(X)=1$ and irregularity $q(X)=dim (X)$ is birational to an abelian variety.

Plucker-Clebsch formula in higher dimension

Let $S\subset\Ps^r$ ($r\geq 5$) be a nondegenerate, irreducible, smooth, complex, projective surface of degree $d$. Let $δ_S$ be the number of double points of a general projection of $S$ to $\Ps^4$. In the present paper we prove that $ δ_S\leq{\binom {d-2} {2}}$, with equality if and only if $S$ is a rational scroll. Extensions to higher dimensions are discussed.

The Hodge number h^1,1 of irregular algebraic surfaces

We prove a new inequality for the Hodge number h^1,1 of irregular complex smooth projective surfaces of general type without irregular pencils of genus >1. More specifically we show that if the irregularity q satisfies q=2^k+1 then h^1,1>4q-4.

Some effects of property $N_p$ on the higher normality and defining equations of nonlinearly normal varieties

For a smooth projective variety $X\subset ¶^r$ embedded by the complete linear system, Property $N_p$ has been studied for a long time. On the other hand, Castelnuovo-Mumford regularity conjecture and related problems have been focused for a projective variety which is not necessarily linearly normal. This paper aims to explain the influence of Property $N_p$ on higher normality and defining equations of a smooth variety embedded by sub-linear system. Also we prove a claim about Property $N_p$ of surface scrolls which is a generalization of Green's work in \cite{Green} about Property $N_p$ for…

特征标圈与超几何族的不规则性Gevrey滤过

Characteristic cycles and the Gevrey filtration of irregularity in hypergeometric families

该研究证明了A-超几何系统特征标圈重数的上半连续性，证实了Schulze与Walther的猜想，还构造了特定半群环揭示Gevrey跳跃轨迹的性质。

Projective Degenerations of K3 Surfaces, Gaussian Maps, and Fano Threefolds

In this article we exhibit certain projective degenerations of smooth $K3$ surfaces of degree $2g-2$ in $\Bbb P^g$ (whose Picard group is generated by the hyperplane class), to a union of two rational normal scrolls, and also to a union of planes. As a consequence we prove that the general hyperplane section of such $K3$ surfaces has a corank one Gaussian map, if $g=11$ or $g\geq 13$. We also prove that the general such hyperplane section lies on a unique $K3$ surface, up to projectivities. Finally we present a new approach to the classification of prime Fano threefolds of index one, which doe…

On the genus of projective curves not contained in hypersurfaces of given degree

Fix integers $r\geq 4$ and $i\geq 2$ (for $r=4$ assume $i\geq 3$). Assuming that the rational number $s$ defined by the equation $\binom{i+1}{2}s+(i+1)=\binom{r+i}{i}$ is an integer, we prove an upper bound for the genus of a reduced and irreducible complex projective curve in $\mathbb P^r$, of degree $d\gg s$, not contained in hypersurfaces of degree $\leq i$. It turns out that this bound coincides with the Castelnuovo's bound for a curve of degree $d$ in $\mathbb P^{s+1}$. We prove that the bound is sharp if and only if there exists an integral surface $S\subset \mathbb P^r$ of degree $s$, n…

Rationally cubic connected manifolds I: manifolds covered by lines

In this paper we study smooth complex projective polarized varieties (X,H) of dimension n \ge 2 which admit a dominating family V of rational curves of H-degree 3, such that two general points of X may be joined by a curve parametrized by V, and such that there is a covering family of rational curves of H-degree one. Our main result is that the Picard number of these manifolds is at most three, and that, if equality holds, (X,H) has an adjuction theoretic scroll structure over a smooth variety.

Derived invariance of the number of holomorphic 1-forms and vector fields

We prove that smooth projective varieties with equivalent derived categories have isogenous (and sometimes isomorphic) Picard varieties. In particular their irregularity and number of independent vector fields are the same. This is turn implies, in combination with the invariance of Hochschild homology, that all Hodge numbers are the same for arbitrary derived equivalent threefolds.

加权射影空间中的极小度数品种

Varieties of minimal degree in weighted projective space

研究加权射影空间中极小度数品种的性质，定义加权1-通用矩阵并探讨其极小度数特征及加权N_p性质，提出关于一般加权三维流形的猜想。

Non-Ulrich representation type

We show that a smooth projective non-degenerate arithmetically Cohen-Macaulay subvariety X of P^N infinite Cohen-Macaulay type becomes of finite Cohen-Macaulay type by removing Ulrich bundles if and only if N = 5 and X is a quartic scroll or the Segre product of a line and a plane. In turn, we give a complete and explicit classification of ACM bundles over these varieties.

A lower bound for $K^2_S$

Let $(S,\mathcal L)$ be a smooth, irreducible, projective, complex surface, polarized by a very ample line bundle $\mathcal L$ of degree $d > 35$. In this paper we prove that $K^2_S\geq -d(d-6)$. The bound is sharp, and $K^2_S=-d(d-6)$ if and only if $d$ is even, the linear system $|H^0(S,\mathcal L)|$ embeds $S$ in a smooth rational normal scroll $T\subset \mathbb P^5$ of dimension $3$, and here, as a divisor, $S$ is linearly equivalent to $\frac{d}{2}Q$, where $Q$ is a quadric on $T$.

皮卡德数为2的光滑射影曲面上的有界上同调性质

Bounded cohomology property on a smooth projective surface with Picard number two

该研究针对皮卡德数为2的光滑射影曲面，在给定闭Mori锥结构的条件下，证明了当满足小平维数不超过1，或小平维数为2且非正则性为0、饭高维数为1时，曲面满足有界上同调性质。

Effective characterization of quasi-abelian surfaces

Let V be a smooth quasi-projective complex surface such that the three first logarithmic plurigenera are equal to 1 and the logarithmic irregularity is equal to 2. We prove that the quasi-Albanese morphism of V is birational and there exists a finite set S such that the quasi-Albanese map is proper over the complement of S in the quasi-Albanese variety A(V) of V. This is a sharp effective version of a classical result of Iitaka.

控制椭圆曲线的射影簇中的非阿基米德整曲线

Non-Archimedean entire curves in projective varieties dominating an elliptic curve

本文针对允许支配态射到椭圆曲线的无群射影曲面，证明其 K-解析 Brody 双曲性，从而建立不规则度为一情形的非阿基米德 Green–Griffiths–Lang 猜想。

Surfaces of general type with q=2 are rigidified

Let $S$ be a minimal smooth projective surface of general type with irregularity $q=2$. We show that, if $S$ has a nontrivial holomorphic automorphism acting trivially on the cohomology with rational coefficients, then it is a surface isogenous to a product. As a consequence of this geometric characterization, one infers that no nontrivial automorphism of surfaces of general type with $q=2$ (which are not necessarily minimal) can be homotopic to the identity. In particular, such surfaces are rigidified in the sense of Fabrizio Catanese.

Cyclic branched coverings of surfaces with abelian quotient singularities

Esnault-Viehweg developed the theory of cyclic branched coverings $\tilde X\to X$ of smooth surfaces providing a very explicit formula for the decomposition of $H^1(\tilde X,\mathbb{C})$ in terms of a resolution of the ramification locus. Later, the first author applies this to the particular case of coverings of $\mathbb{P}^2$ reducing the problem to a combination of global and local conditions on projective curves. In this paper we extend the above results in three directions: first, the theory is extended to surfaces with quotient singularities, second the ramification locus can be partiall…

射影直线覆盖的Tschirnhausen丛

Tschirnhausen bundles of covers of the projective line

研究光滑曲线覆盖射影直线时，结构层前推得到的向量丛的可能类型，对度数为4的情形给出完整答案，并提出一个多面体猜想以刻画本原覆盖的丛，证明所有本原覆盖的丛落在此多面体内且正比例可实现，同时展示本原性假设的必要性及Hurwitz空间像在推广下不保持。

A lower bound for $χ(\mathcal O_S)$

Let $(S,\mathcal L)$ be a smooth, irreducible, projective, complex surface, polarized by a very ample line bundle $\mathcal L$ of degree $d > 25$. In this paper we prove that $χ(\mathcal O_S)\geq -\frac{1}{8}d(d-6)$. The bound is sharp, and $χ(\mathcal O_S)=-\frac{1}{8}d(d-6)$ if and only if $d$ is even, the linear system $|H^0(S,\mathcal L)|$ embeds $S$ in a smooth rational normal scroll $T\subset \mathbb P^5$ of dimension $3$, and here, as a divisor, $S$ is linearly equivalent to $\frac{d}{2}Q$, where $Q$ is a quadric on $T$. Moreover, this is equivalent to the fact that the general hyperpla…

Homaloidal hypersurfaces and hypersurfaces with vanishing Hessian

We prove the existence of various families of irreducible homaloidal hypersurfaces in projective space $\mathbb P^ r$, for all $r\geq 3$. Some of these are families of homaloidal hypersurfaces whose degrees are arbitrarily large as compared to the dimension of the ambient projective space. The existence of such a family solves a question that has naturally arisen from the consideration of the classes of homaloidal hypersurfaces known so far. The result relies on a fine analysis of dual hypersurfaces to certain scroll surfaces. We also introduce an infinite family of determinantal homaloidal hy…

Rank-one sheaves and stable pairs on surfaces

We study rank-one sheaves and stable pairs on a smooth projective complex surface. We obtain an embedding of the moduli space of limit stable pairs into a smooth space. The embedding induces a perfect obstruction theory, which, over a surface with irregularity 0, agrees with the usual deformation-obstruction theory. The perfect obstruction theory defines a virtual fundamental class on the moduli space. Using the embedding, we show that the virtual class equals the Euler class of a vector bundle on the smooth ambient space. As an application, we show that on $\mathbb{P}^2$, the expected count o…

A lower bound for $K_{X}L$ of quasi-polarized surfaces $(X,L)$ with non-negative Kodaira dimension

Let $X$ be a smooth projective surface over the complex number field and let $L$ be a nef-big divisor on $X$. Here we consider the following conjecture; If the Kodaira dimension $κ(X)\geq 0$, then $K_{X}L\geq 2q(X)-4$, where $q(X)$ is the irregularity of $X$. In this paper, we prove that this conjecture is true if (1) the case in which $κ(X)=0$ or 1, (2) the case in which $κ(X)=2$ and $h^{0}(L)\geq 2$, or (3) the case in which $κ(X)=2$, $X$ is minimal, $h^{0}(L)=1$, and $L$ satisfies some conditions.

Intrinsicness of the Newton polygon for smooth curves on $\mathbb{P}^1\times \mathbb{P}^1$

Let $C$ be a smooth projective curve in $\mathbb{P}^1\times \mathbb{P}^1$ of genus $g\neq 4$, and assume that it is birationally equivalent to a curve defined by a Laurent polynomial that is non-degenerate with respect to its Newton polygon $Δ$. Then we show that the convex hull $Δ^{(1)}$ of the interior lattice points of $Δ$ is a standard rectangle, up to a unimodular transformation. Our main auxiliary result, which we believe to be interesting in its own right, is that the first scrollar Betti numbers of $Δ$-non-degenerate curves are encoded in the combinatorics of $Δ^{(1)}$, if $Δ$ satisfie…