Linear systems representing threefolds which are scrolls on a rational surface

Ulrich bundles on three dimensional scrolls

In this paper we construct Ulrich bundles of low rank on three-dimensional scrolls (with respect to the tautological line bundle). We pay special attention to the four types of threefold scrolls in $\mathbb{P}^5$ which were classified in [Ott92].

Abelian surfaces with two plane cubic curve fibrations and Calabi-Yau threefolds

(1,d)-polarized abelian surfaces in P^(d-1) with two plane cubic curve fibrations lie in two elliptic P^2-scrolls. The union of these scrolls form a reducible Calabi-Yau 3-fold. In this paper we show that this occurs when d<10 and analyse the family of such surfaces and 3-folds in detail when d=6. In particular, the reducible Calabi-Yau 3-folds deform in that case to irreducible ones with non-normal singularities.

关于 $\mathbb F_a$ 上某些可分解三维卷丛上的 Ulrich 丛

Ulrich Bundles on decomposable threefold scrolls over $\mathbb F_a$

本文研究 Hirzebruch 曲面上可分解三维卷丛的 Ulrich 丛，完全分类线丛，构造秩二丛并证明其 Ulrich 野性，揭示模空间结构。

某些以$\u211d_e$为底的三维卷轴上的Ulrich丛

Ulrich Bundles on some threefold scrolls over $\mathbb{F}_e$

本研究针对以Hirzebruch曲面$\u211d_e$为底的三维卷轴$X_e$，刻画其上Ulrich向量丛模空间的斜率稳定不可分解分支及其维数与光滑性，算出$X_e$的Ulrich复杂度并证明其几何Ulrich野性，同时推导出$\u211d_e$上向量丛的Ulrich性结果。

Curves with Canonical Models on Scrolls

Let $C$ be an integral and projective curve whose canonical model $C'$ lies on a rational normal scroll $S$ of dimension $n$. We mainly study some properties on $C$, such as gonality and the kind of singularities, in the case where $n=2$ and $C$ is non-Gorenstein, and in the case where $n=3$, the scroll $S$ is smooth, and $C'$ is a local complete intersection inside $S$. We also prove that a rational monomial curve with just one singular point lies on a surface scroll if and only if its gonality is at most $3$, and that it lies on a threefold scroll if and only if its gonality is at most $4$.

The Hilbert curve of a 4-dimensional scroll with a divisorial fiber

In dimension $n = 2m-2 \geq 4$ adjunction theoretic scrolls over a smooth $m$-fold may not be classical scrolls, due to the existence of divisorial fibers. A $4$-dimensional scroll $(X,L)$ over $\mathbb P^3$ of this type is considered, and the equation of its Hilbert curve $Γ$ is determined in two ways, one of which relies on the fact that $(X,L)$ is at the same time a classical scroll over a threefold $Y \not=\mathbb P^3$. It turns out that $Γ$ does not perceive divisorial fibers. The equation we obtain also shows that a question raised in a previous article by Beltrametti, Lanteri and Sommes…

On rationality of nonsingular threefolds with a pencil of Del Pezzo surfaces of degree 4

We prove a criterion of nonsingularity of a complete intersection of two fiberwise quadrics in a scroll over $P^1$. As a corollary we derive the following addition to the Alexeev theorem on rationality of standard Del Pezzo fibrations of degree 4 over $P^1$: we prove that any fibration of this kind with the topological Euler characteristic -4 is rational.

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

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 smooth rational threefolds of P^5 with rational non-special hyperplane section

It is known that the smooth rational threefolds of P^5 having a rational non-special surface of P^4 as general hyperplane section have degree d=3,... ,7. We study such threefolds X from the point of view of linear systems of surfaces in P^3, looking in each case fosr an explicit description of a birational map from P^3 to X. For d=3,..., 6 we prove that there exists a line L on X such that the projection map of X centered at L is birational; we completely describe the base loci B of the linear systems found in this way and give a description of any such threefold X as a suitable blowing-down o…

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…

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…

Rational Curves in Calabi-Yau Threefolds

We study the set of rational curves of a certain topological type in general members of certain families of Calabi-Yau threefolds. For some families we investigate to what extent it is possible to conclude that this set is finite. For other families we investigate whether this set contains at least one point representing an isolated rational curve.

Fano 3-流形的几何 Manin 猜想

Geometric Manin's Conjecture for Fano 3-Folds

本文通过分类光滑 Fano 三维流形上的自由有理曲线族，证明了表示固定数值曲线类的非常自由有理曲线族不可约或为空，从而在三维情形下证明了几何 Manin 猜想，并给出了模空间分量个数的显式计数。