Rational Points on Weighted projective Spaces

An upper bound on the number of rational points of arbitrary projective varieties over finite fields

We give an upper bound on the number of rational points of an arbitrary Zariski closed subset of a projective space over a finite field. This bound depends only on the dimensions and degrees of the irreducible components and holds for very general varieties, even reducible and non equidimensional. As a consequence, we prove a conjecture of Ghorpade and Lachaud on the maximal number of rational points of an equidimensional projective variety.

Low-degree Rational curves on hypersurfaces in projective spaces and their degenerations

We study rational curves on general Fano hypersurfaces in projective space, mostly by degenerating the hypersurface along with its ambient projective space to reducible varieties. We prove results on existence of low-degree rational curves with balanced normal bundle, and reprove some results on irreducibility of spaces of rational curves of low degree.

Existence of rational points on smooth projective varieties

Fix a number field k. We prove that if there is an algorithm for deciding whether a smooth projective geometrically integral k-variety has a k-point, then there is an algorithm for deciding whether an arbitrary k-variety has a k-point and also an algorithm for computing X(k) for any k-variety X for which X(k) is finite. The proof involves the construction of a one-parameter algebraic family of Chatelet surfaces such that exactly one of the surfaces fails to have a k-point.

Classification of embedded projective manifolds swept out by rational homogeneous varieties of codimension one

We give a classification of embedded smooth projective varieties swept out by rational homogeneous varieties whose Picard number and codimension are one.

Holomorphic maps from rational homogeneous spaces onto projective manifolds

Answering a problem raised by Lazarsfeld, Hwang and Mok proved that a surjective holomorphic map from a rational homogeneous space of Picard number 1 onto projective manifold different from projective space must be a biholomorphism. THe aim of this paper is to generalized this result to irreducible rational homogeneous space of higher Picard number.

Rational curves on blowing-ups of projective spaces

We prove the irredcibility (and the rational connectedness) of the moduli spaces of (free) morphisms from a projective line to a successive blowing-up of a product of projective spaces if a suitable numerical condition on morphisms is provided (and also if all centers of the blowing-up are rationally connected).

Fake weighted projective spaces

We define fake weighted projective spaces as a generalisation of weighted projective spaces. We introduce the notions of fundamental group in codimension 1 and of universal covering in codimension 1. We prove that for every fake weighted projective space its universal cover in codimension 1 is a weighted projective space.

Smooth complex projective rational surfaces with infinitely many real forms

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.

A note on rational surfaces in projective four-space

It is proved that a smooth rational surface in projective four-space, which is ruled by cubics or quartics has degree at most 12. It is also proved that a smooth rational surface in projective four-space which is the image of Fn by a linear system with "simple base points" (see text) has degree at most 12.

特征二下平面射影有理四次曲线纤维化

Fibrations by plane projective rational quartic curves in characteristic two

本文在特征二下对平面射影有理四次曲线纤维化进行了完整的双有理等价分类。

Global deformations of certain rational almost homogeneous projective bundles

We study global deformations of certain projective bundles over projective spaces. We show that any projective global deformation of a projective bundle over $\bP^1$ carries the structure of a projective bundle over some projective space. Furthermore, we construct examples in arbitrary dimension $\ge 3$ of almost homogeneous Fano projective bundles over $\bP^2$ which can be globally deformed to non-almost homogeneous manifolds.

加权射影空间与栈的上同调Brauer群

The cohomological Brauer group of weighted projective spaces and stacks

该研究完成了加权射影空间及其栈的扭结构的上同调Brauer群计算，为代数几何相关研究提供了基础结果。

Construction of rational surfaces of degree 12 in projective fourspace (with an appendix by Kristian Ranestad)

The aim of this paper is to present two different constructions of smooth rational surfaces in projective fourspace with degree 12 and sectional genus 13. In particular, we establish the existences of five different families of smooth rational surfaces in projective fourspace with the prescribed invariants.

Computations with rational maps between multi-projective varieties

We shortly describe the algorithms behind some of the functions provided by the Macaulay2 package MultiprojectiveVarieties, a package for multi-projective varieties and rational maps between them.

分类空间的射影逼近的非稳定有理性

Non stable rationality of projective approximations for classifying spaces

本文研究分类空间射影逼近的非稳定有理性，借助 Benoit-Ottem 新不变量与经典非分歧上同调构造了大量反例。

The automorphism group of a rational projective k*-surface

We consider possibly singular rational projective k*-surfaces and provide an explicit description of the unit component of the automorphism group in terms of isotropy group orders and intersection numbers of suitable invariant curves. As an application, we characterize the almost homogeneous rational projective k*-surfaces and we specify the two-dimensional groups acting almost transitively as well as the corresponding homogeneous spaces.

Rationality of cycles on function field of exceptional projective homogeneous varieties

In this article we prove a result comparing rationality of algebraic cycles over the function field of a projective homogeneous variety under a linear algebraic group of type $F_4$ or $E_8$ and over the base field, which can be of any characteristic.

A toric proof of Bézout's theorem for weighted projective spaces

Using toric geometry we prove a Bézout type theorem for weighted projective spaces.

Sharp degree bounds for fake weighted projective spaces

We give sharp upper bounds on the anticanonical degree of fake weighted projective spaces, only depending on the dimension and the Gorenstein index.

A Note on Rational Cuspidal curves on $\mathbb{Q}$-Homology Projective Planes

We generalize results by Wakabayashi and Orevkov about rational cuspidal curves on the projective plane to that on $\mathbb{Q}$-homology projective planes. It turns out that the result is exactly the same as the projective plane case under suitable assumptions. We also provide examples which demonstrate sharpness of the results. The ambient surface is singular in these examples.

The maximum number of singular points on rational homology projective planes

A normal projective complex surface is called a rational homology projective plane if it has the same Betti numbers with the complex projective plane $\mathbb{C}\mathbb{P}^2$. It is known that a rational homology projective plane with quotient singularities has at most 5 singular points. So far all known examples have at most 4 singular points. In this paper, we prove that a rational homology projective plane $S$ with quotient singularities such that $K_S$ is nef has at most 4 singular points except one case. The exceptional case comes from Enriques surfaces with a configuration of 9 smooth ra…

Infinitely generated Lawson homology groups on some rational projective varieties

We construct rational projective 4-dimensional varieties with the property that certain Lawson homology groups tensored with Q are infinite dimensional Q-vector spaces. More generally, each pair of integers p and k, with k\geq 0, p>0, we find a projective variety Y, such that L_pH_{2p+k}(Y) is infinitely generated. We also construct two singular rational projective 3-dimensional varieties Y and Y' with the same homeomorphism type but different Lawson homology groups, specifically L_1H_3(Y) is not isomorphic to L_1H_3(Y')even up to torsion.

Fano hypersurfaces in weighted projective 4-spaces

We determine the complete list of anticanonically embedded quasi smooth log Fano 3-folds in weighted projective 4-spaces. This implies that the Reid-Fletcher list of 95 types of anticanonically embedded quasi smooth terminal Fano threefolds in weighted projective 4-spaces is complete. We also prove that there are only finitely many families of quasi smooth Calabi-Yau hypersurfaces in weighted projective spaces of any given dimension.

On the rationality of the moduli space of instanton bundles on the projective 3-space

We prove the rationality and irreducibility of the moduli space of---what we call---the endomorphism-general instanton vector bundles of arbitrary rank on the projective space. In particular, we deduce the rationality of the moduli spaces of rank-two mathematical instantons. This problem was first studied by Hartshorne, Hirschowitz-Narasimhan in the late 1970s, and it has been reiterated within the framework of the ICM 2018.

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.

真射影相似群的有理连通性

Rational connectedness for groups of proper projective similitudes

本文研究了特征不为2的域上二次型$φ$的真射影相似群${\bf PSim}^+(φ)$的有理连通性，给出了基于$φ$结构性质的充分条件，并构造了非有理连通的新二次型例子。

Enumeration of One-Nodal Rational Curves in Projective Spaces

We give a formula computing the number of one-nodal rational curves that pass through an appropriate collection of constraints in a complex projective space. We combine the methods and results from three different papers.

Computing heights on weighted projective spaces

In this note we extend the concept height on projective spaces to that of weighted height on weighted projective spaces and show how such a height can be computed. We prove some of the basic properties of the weighted height and show how it can be used to study hyperelliptic curves over Q. Some examples are provided from the weighted moduli space of binary sextics and octavics.

Mixed multiplicities and projective degrees of rational maps

We consider the notion of mixed multiplicities for multigraded modules by using Hilbert series, and this is later applied to study the projective degrees of rational maps. We use a general framework to determine the projective degrees of a rational map via a computation of the multiplicity of the saturated special fiber ring. As specific applications, we provide explicit formulas for all the projective degrees of rational maps determined by perfect ideals of height two or by Gorenstein ideals of height three.

On the density of foliations without algebraic solutions on weighted projective planes

We prove that a generic holomorphic foliation on a weighted projective plane has no algebraic solutions when the degree is big enough. We also prove an analogous result for foliations on Hirzebruch surfaces.

Compactified moduli spaces of rational curves in projective homogeneous varieties

The space of smooth rational curves of degree $d$ in a projective variety $X$ has compactifications by taking closures in the Hilbert scheme, the moduli space of stable sheaves or the moduli space of stable maps respectively. In this paper we compare these compactifications by explicit blow-ups and -downs when $X$ is a projective homogeneous variety and $d\leq 3$. Using the comparison result, we calculate the Betti numbers of the compactifications when $X$ is a Grassmannian variety.

Rational curves in CICY's in products of two projective spaces

Let $X$ be the product of two projective spaces and consider the general CICY threefold $Y$ in $X$ with configuration matrix $A$. We prove the finiteness part of the analogue of the Clemens' conjecture for such a CICY in low bidegrees. More precisely, we prove that the number of smooth rational curves on $Y$ with low bidegree and with nondegenerate birational projection is at most finite (even in cases in which positive dimensional families of degenerate rational curves are known).

Stable rationality of del Pezzo fibrations of low degree over projective spaces

The main aim of this article is to show that a very general 3-dimensional del Pezzo fibration of degree 1,2,3 is not stably rational except for a del Pezzo fibration of degree 3 belonging to explicitly described 2 families. Higher dimensional generalizations are also discussed and we prove that a very general del Pezzo fibration of degree 1,2,3 defined over the projective space is not stably rational provided that the anticanonical divisor is not ample.

有理加权射影超曲面

Rational weighted projective hypersurfaces

本文研究了加权射影超曲面的有理性，引入了高次加权超曲面的有理性构造，提供了任意域上的新有理例子，并肯定回答了T. Okada关于所有维数$n \geq 6$下存在非常一般端Fano有理加权超曲面的问题。

The Number of Rational Quartics on Calabi-Yau hypersurfaces in Weighted Projective Space P(2,1^4)

We compute the number of rational quartics on a general Calabi-Yau hypersurface in weighted projective space P(2,1^4). The result agrees with the prediction made by mirror symmetry.

射影超曲面的 Hirzebruch-Milnor 类检测更高 du Bois 或有理奇点的例子

Examples of Hirzebruch-Milnor classes of projective hypersurfaces detecting higher du Bois or rational singularities

本文表明经典 Hirzebruch-Milnor 类仅在特殊情形下能检测射影超曲面的更高 du Bois 或有理奇点，并澄清了作者早期论文中的若干要点。

A connectedness theorem for products of weighted projective spaces

We prove a connectedness result for products of weighted projective spaces.

关于主导有理自映射在射影簇上周期点和前周期点高度有界的性质

On the height boundedness of periodic and preperiodic points of dominant rational self-maps on projective varieties

本文推翻了关于周期点高度有界的猜想，并证明了在射影簇上某些主导有理自映射的周期点集高度有界，同时指出前周期点可能不满足这一性质。

Hori--Vafa mirror models for complete intersections in weighted projective spaces and weak Landau--Ginzburg models

We prove that Hori--Vafa mirror models for smooth Fano complete intersections in weighted projective spaces admit an interpretation as Laurent polynomials.

Counting rational curves of arbitrary shape in projective spaces

We present an approach to a large class of enumerative problems concerning rational curves in projective spaces. This approach uses analysis to obtain topological information about moduli spaces of stable maps. We demonstrate it by enumerating one-component rational curves with a triple point or a tacnodal point in the three-dimensional projective space and with a cusp in any projective space.

Exterior algebra methods for the construction of rational surfaces in the projective fourspace

The aim of this paper is to present a construction of smooth rational surfaces in projective fourspace with degree 12 and sectional genus 13. The construction is based on exterior algebra methods, finite field searches and standard deformation theory.

Mirror fibrations and root stacks of weighted projective spaces

We show that the orbifold Chow ring of a root stack over a well-formed weighted projective space can be naturally seen as the Jacobian algebra of a function on a singular variety given by a partial compactification of its Ginzburg-Landau model.

Kahler-Einstein metrics on log del Pezzo surfaces in weighted projective 3-spaces

We determine the complete list of anticanonically embedded quasi smooth log del Pezzo surfaces in weighted projective 3-spaces. We prove that many of these admit a Kähler-Einstein metric and most of them do not have tigers.

Smooth rational projective varieties with non-finitely generated discrete automorphism group and infinitely many real forms

We show, among other things, that for each integer $n \ge 3$, there is a smooth complex projective rational variety of dimension $n$, with discrete non-finitely generated automorphism group and with infinitely many mutually non-isomorphic real forms. Our result is inspired by the work of Lesieutre and the work of Dinh and Oguiso.

Computational procedures for weighted projective spaces

This is a pdf print of the homonymous Maple file, freely available at http://www.maplesoft.com/applications/view.aspx?SID=127621, providing procedures which are able to produce the toric data associated with a (polarized) weighted projective space i.e. fans, polytopes and their equivalences. More originally it provides procedures which are able to detect a weights vector Q starting from either a fan or a polytope: we will call this process the recognition of a (polarized) weighted projective space. Moreover it gives procedures connecting polytopes of a polarized weighted projective space with …

Constructing projective varieties in weighted flag varieties

We compute the Hilbert series of general weighted flag varieties and discuss a computer-aided method to determine their defining equations. We apply our results to weighted flag varieties coming from the Lie groups of type G_2 and GL(6), to construct some families of polarised projective varieties in codimensions 8 and 6, respectively.

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.

$\mathbb{P}^{n}$ 的加权射影退化

Weighted projective degenerations of $\mathbb{P}^{n}$

研究射影空间 $\mathbb{P}^{n}$ 的加权射影 klt $\mathbb{Q}$-Gorenstein 退化，构造了任意维度的无穷多个新退化，并详细研究了任意加权射影三维簇的形变。

Kac's Theorem for weighted projective lines

We prove an analogue of Kac's Theorem, describing the dimension vectors of indecomposable coherent sheaves, or parabolic bundles, over weighted projective lines. We use a theorem of Peng and Xiao to associate a Lie algebra to the category of coherent sheaves for a weighted projective line over a finite field, and find elements of this Lie algebra which satisfy the relations defining the loop algebra of a Kac-Moody Lie algebra.

Gauss-Manin systems of wild regular functions: Givental-Hori-Vafa models of smooth hypersurfaces in weighted projective spaces as an example

We study Gauss-Manin systems of non tame Laurent polynomial functions. We focuse on Hori-Vafa models, which are the expected mirror partners of the small quantum cohomology of smooth hypersurfaces in weighted projective spaces.

偏极化圆柱在加权射影平面吹胀表面中的研究

Polarized cylinders on blow-ups of weighted projective planes

研究加权射影平面吹胀得到的有理表面中的偏极化圆柱，分析其作为加权超曲面或拟光滑完全交集的自然出现形式。

加权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 weighted projective spaces

本文研究加权射影空间中非常一般超曲面的代数双曲性，通过给出丰沛线丛倍数 $m$ 的界，证明了相应零点轨迹具有代数双曲性。

Construction of bicuspidal rational complex projective plane curves

We give several constructions of bicuspidal rational complex projective plane curves, and list the Newton pairs and the multiplicity sequences of the singularities on the resulting curves. Although the existence of some of the listed cusp configurations is already proved in the literature, the advantage of our approach is that on one hand we explicitly state the Newton pairs and the multiplicity sequences of the singularities on the curves, on the other hand, we also give birational transformations transforming a conic or a bicuspidal curve with simple equation to these rational bicuspidal cur…

Weighted projective lines as fine moduli spaces of quiver representations

We describe weighted projective lines in the sense of Geigle and Lenzing by a moduli problem on the canonical algebra of Ringel. We then go on to study generators of the derived categories of coherent sheaves on the total spaces of their canonical bundles, and show that they are rarely tilting. We also give a moduli construction for these total spaces for weighted projective lines with three orbifold points.