On normal K3 surfaces

有理齐性空间中的正常分裂除子

Normal split divisors in rational homogeneous spaces

研究有理齐性空间中具有分裂正常序列的除子性质，揭示其与投影空间或二次曲面超平面截面的关系。

Irreducible Components of Hilbert Schemes of Rational Curves with given Normal Bundle

We develop a new general method for computing the decomposition type of the normal bundle to a projective rational curve. This method is then used to detect and explain an example of a Hilbert scheme that parametrizes all the rational curves in $\mathbb{P}^s$ with a given decomposition type of the normal bundle and that has exactly two irreducible components. This gives a negative answer to the very old question whether such Hilbert schemes are always irreducible. We also characterize smooth non-degenerate rational curves contained in rational normal scroll surfaces in terms of the splitting t…

具有有理同伦圆盘光滑化的正规曲面奇点

Normal complex surface singularities with rational homology disk smoothings

本文证明极小好预解图至少含两个节点的正规曲面奇点无有理同伦圆盘光滑化，结合已有分类结果给出此类奇点的完整分类，同时证明了Wahl猜想及其有理猜想。

On rational normal curves in projective space

In this paper we consider a generalization of a well known result by Veronese about rational normal curves. More precisely, given a collection of linear spaces in $\PP^n$ we study the existence of rational normal curves intersecting each component of the configuration maximally. We introduce different methods to show existence and non-existence of such curves. We also show how to apply these techniques to the study of defectivity of Segre-Veronese varieties.

Coverings of rational ruled normal surfaces

In this work we use arithmetic, geometric, and combinatorial techniques to compute the cohomology of Weil divisors of a special class of normal surfaces, the so-called rational ruled toric surfaces. These computations are used to study the topology of cyclic coverings of such surfaces ramified along Q-normal crossing divisors.

Normal bundles of rational curves in projective spaces

We determine the splitting (isomorphism) type of the normal bundle of a generic genus-0 curve with 1 or 2 components in any projective space, as well as the (sometimes nontrivial) way the bundle deforms locally with a general deformation of the curve. We deduce an enumerative formula for divisorial loci of smooth rational curves whose normal bundle is of non-generic splitting type.

Galois subspaces for the rational normal curve

We characterize all $(n-2)$-dimensional linear subspaces of $\mathbb{P}^{n}$ such that the induced linear projection, when restricted to the rational normal curve, gives a Galois morphism. We give an explicit description of these spaces as a disjoint union of locally closed subvarieties in the Grassmannian $\mathbb{G}(n-2,n)$.

有理正规涡卷面上对数丛的Castelnuovo-Mumford正则性与分裂判据

Castelnuovo-Mumford Regularity and Splitting Criteria for Logarithmic Bundles over Rational Normal Scroll Surfaces

本文在有理正规涡卷面上引入Castelnuovo-Mumford正则性，证明凝聚层分裂判据与Ulrich丛刻画，并研究直线和有理曲线配置的对数丛。

On linear systems with multiple points on a rational normal curve

We give a closed formula for the dimension of all linear systems in $\mathbb{P}^n$ with assigned multiplicity at arbitrary collections of points lying on a rational normal curve of degree $n$. In particular we give a purely geometric explanation of the speciality of these linear systems, which is due to the presence of certain subvarieties in the base locus: linear spans of points, secant varieties of the rational normal curve or joins between them.

Normal bundles of rational curves on complete intersections

Let $X \subset \mathbb{P}^n$ be a general Fano complete intersection of type $(d_1,\dots, d_k)$. If at least one $d_i$ is greater than $2$, we show that $X$ contains rational curves of degree $e \leq n$ with balanced normal bundle. If all $d_i$ are $2$ and $n\geq 2k+1$, we show that $X$ contains rational curves of degree $e \leq n-1$ with balanced normal bundle. As an application, we prove a stronger version of the theorem of Z. Tian \cite{Tian}, Q. Chen and Y. Zhu \cite{ChenZhu} that $X$ is separably rationally connected by exhibiting very free rational curves in $X$ of optimal degrees.

Normal Bundle of Rational Curves and Waring Decomposition

The problem of determining the splitting of the normal bundle of rational space curves has been considered in the 80s in a series of papers by Ghione and Sacchiero and by Eisenbud and Van de Ven. With our approach we are able to obtain results for curves embedded in $\bbP^m$ for $m\geq 3$ and we find an interesting interplay between the Waring decomposition of binary forms and the splitting of the normal bundle.

A note on rational normal curves totally tangent to a Hermitian variety

Let q be a power of a prime integer p, and let X be a Hermitian variety of degree q+1 in the n-dimensional projective space. We count the number of rational normal curves that are tangent to X at distinct q+1 points with intersection multiplicity n. This generalizes a result of B. Segre on the permutable pairs of a Hermitian curve and a smooth conic.

Rational and Semi-Rational Singularities

It is a conjecture of Kollár that a variety $X$ with rational singularities in some open subvariety $U$ has a rationalification; that is, a proper, birational morphism $f: Y \rightarrow X$ such that $Y$ has rational singularities, and which is an isomorphism over $U$. Whether this is true is already unknown in the case of a (normal) threefold with rational singularities along a curve $C$ except at a single point $p \in C$. There is an analogous conjecture for demi-normal varieties $X$, where we must insist that $Y$ has only semi-rational singularities. Our main result is that if a stronger ver…

Demi-Normal Surface Singularities

We prove semi-rationalification and semi-log-canonicalization for Gorenstein demi-normal surfaces. That is, given a Gorenstein demi-normal surface X with semi-rational (respectively, semi-log canonical) singularities in an open set U with complement a finite set of points, there is a proper birational morphism f : Y --> X such that f is an isomorphism over U and Y has only semi-rational (respectively, semi-log canonical) singularities. We proceed by passing to the normalization and then gluing along the conductor in an appropriate rationalification or log-canonicalization of the normalization …

Stable rationality of Brauer-Severi surface bundles

For sufficiently ample linear systems on rational surfaces we show that a very general associated Brauer-Severi surface bundle is not stably rational.

有理三维流形上Brauer-Severi曲面丛的有理性

Rationality of Brauer-Severi surface bundles over rational 3-folds

本文研究有理三维流形上Brauer-Severi曲面丛的稳定有理性问题，给出了其不是稳定有理的一个充分条件，并通过具体例子和构造一族一般成员光滑且非稳定有理的曲面丛加以验证。

Strong Rational Connectedness of Surfaces

We discuss the strong rational connectedness of smooth rationally connected surfaces. We prove in lots of cases, including the smooth locus of a log del Pezzo surface, the rational connectedness indeed implies the strong rational conectedness. This confirms a conjecture due to Hassett and Tschinkel in \cite{ht08}.

Smooth rational curves on rational surfaces

Consider the scheme parametrizing non-constant morphisms from a fixed projective curve to a projective surface. There is a rational map between this scheme and the Chow variety of $1$-cycles on the surface. We prove that, if the curve is non-singular, then this rational map is a morphism. As a consequence, we obtain that, if the surface is rational and we fix a divisor class containing a non-singular rational curve, then the scheme parametrizing rational curves on this class is irreducible. Further, if the class has non-negative self-intersection, then the scheme of rational curves has expecte…

Very free rational curves in Fano varieties

Let $X$ be a projective variety and let $C$ be a rational normal curve on $X$. We compute the normal bundle of $C$ in a general complete intersection of hypersurfaces of sufficiently large degree in $X$. As a result, we establish the separable rational connectedness of a large class of varieties, including general Fano complete intersections of hypersurfaces of degree at least three in flag varieties, in arbitrary characteristic. In addition, we give a new way of computing the normal bundle of certain rational curves in products of varieties in terms of their restricted tangent bundles and nor…

An effective criterion for algebraicity of rational normal surfaces

We give a novel and effective criterion for algebraicity of rational normal analytic surfaces constructed from resolving the singularity of an irreducible curve-germ on $CP^2$ and contracting the strict transform of a given line and all but the `last' of the exceptional divisors. As a by-product we construct a new class of analytic non-algebraic rational normal surfaces which are `very close' to being algebraic. These results are local reformulations of some results in (Mondal, 2011) which sets up a correspondence between normal algebraic compactifications of $C^2$ with one irreducible curve a…

Painlevé equations and deformations of rational surfaces with rational double points

In this paper, we show that the Bäcklund transformations of Painlevé equations come from birational maps of rational surfaces constructed by Okamoto as the spaces of initial conditions. The simultaneous resolutions of rational double points of the families of rational surfaces give rise to flops which are origins of symmetry groups of Painlevé equation. Moreover we point out that Kodaira--Spencer theory of deformations of rational surfaces explains a geomrtric meaning of Painlevé equations.

Linear systems on rational elliptic surfaces and elliptic fibrations on K3 surfaces

We consider K3 surfaces which are double cover of rational elliptic surfaces. The former are endowed with a natural elliptic fibration, which is induced by the latter. There are also other elliptic fibrations on such K3 surfaces, which are necessarily induced by special linear systems on the rational elliptic surfaces. We describe these linear systems. In particular, we observe that every conic bundle on the rational surface induces a genus 1 fibration on the K3 surface and we classify the singular fibers of the genus 1 fibration on the K3 surface it terms of singular fibers and special curves…

Counting real rational curves on K3 surfaces

We provide a real analog of the Yau-Zaslow formula counting rational curves on $K3$ surfaces.

Rational Curves on K3 Surfaces

We proved the existence of rational curves in every linear system on a general K3 surface and that all rational curves in the hyperplane class are nodal on a general K3 surface of small genus.

Smooth Rational Curves on Singular Rational Surfaces

We classify all complex surfaces with quotient singularities that do not contain any smooth rational curves, under the assumption that the canonical divisor of the surface is not pseudo-effective. As a corollary we show that if $X$ is a log del Pezzo surface such that for every closed point $p\in X$, there is a smooth curve (locally analytically) passing through $p$, then $X$ contains at least one smooth rational curve.

$G$-solid 有理曲面

$G$-solid Rational Surfaces

本文在复数域上分类了 $G$-solid 有理曲面，给出了完整的分类结果。

On the projective normality of double coverings over a rational surface

We study the projective normality of a minimal surface $X$ which is a ramified double covering over a rational surface $S$ with $\dim|-K_S|\ge 1$. In particular Horikawa surfaces, the minimal surfaces of general type with $K^2_X=2p_g(X)-4$, are of this type, up to resolution of singularities. Let $π$ be the covering map from $X$ to $S$. We show that the $\mathbb{Z}_2$-invariant adjoint divisors $K_X+rπ^*A$ are normally generated, where the integer $r\ge 3$ and $A$ is an ample divisor on $S$.

Rational Points on Del Pezzo Surface of degree 1 and 2

We discuss the rational points on del Pezzo surface of degree 1 and 2 over any finite field $\mathbb F_q$, and give out the explicit equations of del Pezzo surfaces that have unique rational point.

Self Rational Maps of K3 Surfaces

We prove that a very general projective K3 surface does not admit a dominant self rational map of degree at least two.

Singularities of Rational Curves on K3 surfaces

We proved that every rational curves in the primitive class of a general K3 surface of any genus is nodal.

Cylinders in rational surfaces

We answer a question of Ciro Ciliberto about cylinders in rational surfaces which are obtained by blowing up the plane at points in general position.

Kawamata-Viehweg Vanishing on Rational Surfaces in Positive Characteristic

We prove that the Kawamata-Viehweg vanishing theorem holds on rational surfaces in positive characteristic by means of the lifting property to W_2(k) of certain log pairs on smooth rational surfaces. As a corollary, the Kawamata-Viehweg vanishing theorem holds on log del Pezzo surfaces in positive characteristic.

Potentially stably rational del Pezzo surfaces over nonclosed fields

We classify Galois actions on Picard lattices of del Pezzo surfaces of degrees 1,2, and 3 giving rise to minimal surfaces with no cohomological obstructions to stable rationality.

Rank-3 stable bundles on rational ruled surfaces

In this paper, we compare the moduli spaces of rank-3 vector bundles stable with respect to different ample divisors over rational ruled surfaces. We also discuss the irreducibility, unirationality, and rationality of these moduli spaces.

On the saturation sequence of the rational normal curve

Let $C \subseteq ¶^d$ denote the rational normal curve of order $d$. Its homogeneous defining ideal $I_C \subseteq \QQ[a_0,...,a_d]$ admits an $SL_2$-stable filtration $J_2 \subseteq J_4 \subseteq ... \subseteq I_C$ by sub-ideals such that the saturation of each $J_{2q}$ equals $I_C$. Hence, one can associate to $d$ a sequence of integers $(α_1,α_2,...)$ which encodes the degrees in which the successive inclusions in this filtration become trivial. In this paper we establish several lower and upper bounds on the $α_q$, using \emph{inter alia} the methods of classical invariant theory.

Rational curves and points on K3 surfaces

We study the distribution of algebraic points on K3 surfaces.

Smooth rational surfaces violating Kawamata-Viehweg vanishing

We show that over any algebraically closed field of positive characteristic, there exists a smooth rational surface which violates Kawamata-Viehweg vanishing.

Potential density of rational points for K3 surfaces over function fields

We give examples of non-isotrivial K3 surfaces over complex function fields with Zariski-dense rational points and N'eron-Severi rank one.

A categorical invariant for geometrically rational surfaces with a conic bundle structure

We define a categorical birational invariant for minimal geometrically rational surfaces with a conic bundle structure over a perfect field via components of a natural semiorthogonal decomposition. Together with the similar known result on del Pezzo surfaces, this provides a categorical birational invariant for geometrically rational surfaces.

Grothendieck-Lefschetz Theory, Set-Theoretic Complete Intersections and Rational Normal Scrolls

Using the Grothendieck-Lefschetz theory (see \cite{[SGA2]}) we prove a criterion to deduce that certain subvarieties of $\mathbb P^n$ of dimension $\geq 2$ are not set-theoretic complete intersections (see Theorem 1 of the Introduction). As applications we give a number of relevant examples. In the last part of the paper we prove that the arithmetic rank of a rational normal $d$-dimensional scroll $S_{n_1,...,n_d}$ in $\mathbb P^N$ is $N-2$, by producing an explicit set of $N-2$ homogeneous equations which define these scrolls set-theoretically (see Theorem 2 of the Introduction).

实有理曲面上的曲线逼近

Approximating curves on real rational surfaces

本文给出实有理曲面上简单闭曲线可被光滑有理曲线逼近的拓扑充要条件，并研究了给定复自交数的逼近问题。

Grassmannians 中有理曲线法丛的平衡性

Balancedness of Normal Bundles of Rational Curves in Grassmannians

本文证明了 Coskun-Larson-Vogt 关于 Grassmannians 中有理曲线法丛平衡性的猜想在 G(2,4)、G(2,5) 和 G(2,6) 上成立，推广了射影空间中的平衡性结果。

On rational maps from a general surface in P^3 to surfaces of general type

We study the dominant rational maps from a general surface in P^{3} to surfaces of general type. We prove restrictions on the target surfaces, and special properties of the rational maps. We show that for a small degree the general surface has no such map. Moreover a slight improvement of a result of Catanese, on the number of moduli of a surface of general type, is also obtained.

The Hilbert schemes of points on surfaces with rational double point singularities

We prove that the Hilbert scheme of points on a normal quasi-projective surface with at worst rational double point singularities is irreducible.

The irreducibility of the spaces of rational curves on del Pezzo surfaces

We prove that the spaces of rational curves on del Pezzo surfaces are either irreducible or empty, with a unique exception.

Webs of rational curves on real surfaces and a classification of real weak del Pezzo surfaces

We classify webs of minimal degree rational curves on surfaces and give a criterion for webs being hexagonal. In addition, we classify Neron-Severi lattices of real weak del Pezzo surfaces. These two classifications are related to root subsystems of E8.

Unbounded negativity on rational surfaces in positive characteristic

We give explicit blowups of the projective plane in positive characteristic that contain smooth rational curves of arbitrarily negative self-intersection, showing that the Bounded Negativity Conjecture fails even for rational surfaces in positive characteristic.

Alternating groups and rational functions on surfaces

Let X be a smooth complex projective surface. We prove that for any sufficiently big m there exists a rational dominant map f from X into a complex rational ruled surface Y, such that f is generically finite of degree m and has monodromy the alternating group Am.

Linear systems of rational curves on rational surfaces

Given a curve C on a projective nonsingular rational surface S, over an algebraically closed field of characteristic zero, we are interested in the set Omega_C of linear systems Lambda on S satisfying C is in Lambda, dim Lambda > 0, and the general member of Lambda is a rational curve. The main result of the paper gives a complete description of Omega_C and, in particular, characterizes the curves C for which Omega_C is non empty.

Rational double points on Enriques surfaces

We classify, up to some lattice-theoretic equivalence, all possible configurations of rational double points that can appear on a surface whose minimal resolution is a complex Enriques surface.

一个具有不可数多个实形式的光滑复有理仿射曲面

A smooth complex rational affine surface with uncountably many real forms

本文构造了一个光滑复有理仿射曲面，证明其具有不可数多个互不同构的实形式，揭示了实代数几何中实形式的丰富性与复杂性。

Real forms on rational surfaces

For any positive integer $r$, we construct a smooth complex projective rational surface which has at least $r$ real forms not isomorphic over $\mathbb{R}$.

Hilbert schemes of two points on K3 surfaces and certain rational cubic fourfolds

In this paper, we check that Fano schemes of lines on certain rational cubic fourfolds are birational to Hilbert schemes of two points on K3 surfaces.

Contour curves and isophotes on rational ruled surfaces

The ruled surfaces, i.e., surfaces generated by one parametric set of lines, are widely used in the~field of applied geometry. An~isophote on a surface is a curve consisting of surface points whose normals form a constant angle with some fixed vector. Choosing an angle equal to $π/2$ we obtain a special instance of a~isophote -- the so called contour curve. While contours on rational ruled surfaces are rational curves, this is no longer true for the isophotes. Hence we will provide a formula for their genus. Moreover we will show that the only surfaces with a~rational generic contour are just …