A connectedness theorem for products of weighted projective spaces

On Rational Connectedness of Globally F-Regular Threefolds

In this paper, we show that projective globally $F$-regular threefolds, defined over an algebraically closed field of characteristic $p\geq 11$, are rationally chain connected.

Regulating Hartshorne's connectedness theorem

A classical theorem by Hartshorne states that the dual graph of any arithmetically Cohen--Macaulay projective scheme is connected. We give a quantitative version of Hartshorne's result, in terms of Castelnuovo--Mumford regularity. If $X \subset \mathbb{P}^n$ is an arithmetically Gorenstein projective scheme of regularity $r+1$, and if every irreducible component of $X$ has regularity $\le r'$, we show that the dual graph of $X$ is $\lfloor{\frac{r+r'-1}{r'}}\rfloor$-connected. The bound is sharp. We also provide a strong converse to Hartshorne's result: Every connected graph is the dual graph …

特征零域上的单有理性与有理连通性等价

Unirationality is the same thing as Rational Connectedness in Characteristic Zero

本文证明特征零域上光滑射影变体的单有理性、有理连通性与有理链连通性等价，通过引入MU(X)并结合MRC纤维丛与归纳论证完成证明。

Campana 可分离有理连通性 of 纯群轨道折叠

Campana separable rational connectedness of toric orbifold

本文证明了光滑非klt轨道折叠具有可分离Campana有理连通性，并展示了在正特征下，某些奇异加权射影空间不满足此性质。

Connectedness Bertini Theorem via numerical equivalence

Let $X$ be an irreducible projective variety and $f$ a morphism $X \rightarrow \mathbb{P}^n$. We give a new proof of the fact that the preimage of any linear variety of dimension $k\ge n+1-\dim f(X)$ is connected. We prove that the statement is a consequence of the Generalized Hodge Index Theorem using easy numerical arguments that hold in any characteristic. We also prove the connectedness Theorem of Fulton and Hansen as application of our main theorem.

Stability conditions on curves

We study some examples of Bridgeland-Douglas stability conditions on triangulated categories. From one side we give a complete description of the stability manifolds for smooth projective curves of positive genus. From the other side we study stability conditions on triangulated categories generated by an exceptional collection. In the case of the projective line this leads to the connectedness and simply-connectedness of the stability manifold.

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).

赋值环空间的连通性定理

A connectedness theorem for spaces of valuation rings

本文将 Zariski 连通性定理提升至赋值环的 Zariski-Riemann 空间，证明子环为局部、剩余代数且整闭当且仅当其是某闭连通子空间中环的交集，并给出该定理的逆命题，且无需 Noetherian 假设。

$\u7b49\u6570\u66f2\u7ebf\u4e0a\u77e2\u91cf\u7ebf\u7f1d\u6a21\u79c0\u7684$\mathbb{A}^1$-连通性

$\mathbb{A}^1$-connectedness of moduli of vector bundles on a stacky curve

本文证明了无限域上栈式曲线具有固定行列式和重数的$\mathcal{E}$-半稳定向量丛模栈的$\mathbb{A}^1$-连通性，并推论出抛物$\alpha$-半稳定向量丛模栈的$\mathbb{A}^1$-连通性。

The Realizable Extension Problem and the Weighted Graph $(K_{3,3},l)$

This note outlines the realizable extension problem for weighted graphs and provides results of a detailed analysis of this problem for the weighted graph $(K_{3,3},l)$. This analysis is then utilized to provide a result relating to the connectedness of the moduli space of planar realizations of $(K_{3,3},l)$. The note culminates with two examples which show that in general, realizability and connectedness results relating to the moduli spaces of weighted cycles which are contained in a larger weighted graph cannot be extended to similar results regarding the moduli space of the larger weighte…

Balanced rational curves and minimal rational connectedness of Fano hypersurfaces

On a general Fano hypersurface in projective space, we determine for infinitely many $k$ the minimal degree $e$ of a rational curve through a general collection of $k$ points. In the case of a hypersurface of index 1, our results hold for all $k\geq 1$. In an appendix, M.C. Chang proves an arithmetical result which implies that in the case of index $>1$, the density of the set of curve degrees $e$ covered by our method is approximately $\frac{(n-d)(d-\frac{5}{2})}{(n-2)d}$.

Rational connectedness and Galois covers of the projective line

Let k be a p-adic field. Some time ago, D. Harbater [9] proved that any finite group G may be realized as a regular Galois group over the rational function field in one variable k(t), namely there exists a finite field extension $F/k(t)$, Galois with group G, such that F is a regular extension of k (i.e. k is algebraically closed in F). Moreover, one may arrange that a given k-place of k(t) be totally split in F. Harbater proved this theorem for k an arbitrary complete valued field. Rather formal arguments ([10, §4.5]; \S2 hereafter) then imply that the theorem holds over any `large' field k. …

加权射影空间中的有理正规曲线

Rational Normal Curves in Weighted Projective Space

该研究将有理正规曲线的经典同调理论推广至加权射影空间，建立了行列式性以及二次生成与 Koszul 性质的非标准形式。

加权射影空间中曲线的线性合冲

Linear syzygies of curves in weighted projective space

本文将Green的$N_p$条件推广到加权射影空间，证明曲线高次嵌入满足这些条件，并在广泛情形下建立了正性与低次虚拟合冲之间的联系。

Embeddings of C*-surfaces into weighted projective spaces

Let V be a normal affine surface which admits a C*- and a C+-action. In this note we show that in many cases V can be embedded as a principal Zariski open subset into a hypersurface of a weighted projective space. In particular, we recover a result of D. Daigle and P. Russell.

加权射影空间的渐近syzygies

Asymptotic Syzygies of Weighted Projective Spaces

本文通过改进Ein-Erman-Lazarsfeld的方法，证明了在加权射影空间$\mathbb{P}(1^n,2)$中关于Veronese嵌入渐近syzygies的类比结果。

Homogeneous Fibrations over Curves

By studying the theory of rational curves, we introduce a notion of rational simple connectedness for projective homogeneous spaces. As an application, we prove that over a function field of an algebraic surface, a projective homogeneous space admits a rational point if the elementary obstruction vanishes.

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。

Codes over a weighted torus

We define weighted projective Reed-Muller codes over a subset of weighted projective space over a finite field. We focus on the case when the set X is a projective weighted torus. We show that the vanishing ideal of X is a lattice ideal and relate it with the lattice ideal of a minimal presentation of the semigroup algebra of Q, the numerical semigroup generated by the weights of the projective space. We compute the index of regularity of the vanishing ideal as function of the weights and the Frobenius number of Q. We compute the basic parameters of weighted projective Reed-Muller codes over a…

实闭域上的广义连通性与Bertini型定理

Generalized connectedness and Bertini-type theorems over real closed fields

本文建立了Bertini定理在实闭域上的类比，证明了可逆层截面的零轨迹若具形式实泛点则不变号，并证明维数≥2的拟射影簇存在保持形式实性与整性的超曲面截面非空开子集。

加权射影空间中的插值

Interpolation in Weighted Projective Spaces

本文利用交换代数方法研究加权射影空间中的双重插值问题，证明一般简单点的 Hilbert 函数具有预期行为，并通过归纳程序和 Terracini 引理给出 Alexander-Hirschowitz 定理在加权射影平面中的无例外情形及无穷族插值下界。

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.

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。

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

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

The Integer Decomposition Property and Weighted Projective Space Simplices

Reflexive lattice polytopes play a key role in combinatorics, algebraic geometry, physics, and other areas. One important class of lattice polytopes are lattice simplices defining weighted projective spaces. We investigate the question of when a reflexive weighted projective space simplex has the integer decomposition property. We provide a complete classification of reflexive weighted projective space simplices having the integer decomposition property for the case when there are at most three distinct non-unit weights, and conjecture a general classification for an arbitrary number of distin…

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.

On the Connectedness of the Moduli Space of Calabi--Yau Manifolds

We show that the moduli space of all Calabi-Yau manifolds that can be realized as hypersurfaces described by a transverse polynomial in a four dimensional weighted projective space, is connected. This is achieved by exploiting techniques of toric geometry and the construction of Batyrev that relate Calabi-Yau manifolds to reflexive polyhedra. Taken together with the previously known fact that the moduli space of all CICY's is connected, and is moreover connected to the moduli space of the present class of Calabi-Yau manifolds (since the quintic threefold P_4[5] is both CICY and a hypersurface …

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.

消散定理与全纯张量场的有理连通性

Vanishing theorems and rational connectedness on holomorphic tensor fields

本文通过建立统一RC k-正的全纯向量丛的消散定理，证明了具有正k-瑞奇曲率的紧复流形的全纯切丛是统一RC k-正的，并应用该结果证明某些全纯张量场空间的平凡性及紧凯勒流形的有理连通性。

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.

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.

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.

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.

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.

$\mathbb{P}^{n}$ 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。

加权射影平面上码的代数不变量

Algebraic Invariants of Codes on Weighted Projective Planes

本文利用交换代数与同调代数方法研究$\mathbb{P}(1,1,a)$上的WPRM码，通过极小自由分解得到有理点消失理想的Hilbert级数并确定码参数。

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.

加权射影环面子群上的码

Codes on Subgroups of Weighted Projective Tori

本文研究加权射影环面子群上的码，通过推导相关代数不变量，完整计算了 P(1,1,a) 加权射影平面中此类子群上广义环面码的主要参数。

On the derived category of a weighted projective threefold

We calculate a semi-orthogonal decomposition of the bounded derived category of coherent sheaves on P(1,1,1,3) using a tilting bundle.

加权射影超曲面的自同构

Automorphisms of weighted projective hypersurfaces

本文研究拟光滑复加权射影超曲面的自同构群，证明自同构在多数情形下可延拓到外围空间，给出线性自同构群有限的刻画及一致上界，并描述了给定权和次数的通用超曲面的自同构。

Desingularizations of some Weighted Projective Planes

In this paper we discuss the desingularization algorithm for a toric surface. In particular, we construct an iterable method of determining the Hirzebruch-Jung continued fraction decomposition. These results are then applied to weighted projective planes with at least one tivial weight, ${\mathbb P}(1,m,n)$. The paper concludes with the development of a computer program that computes this continued fraction decomposition.

Classifying terminal weighted projective space

We present a classification of all weighted projective spaces with at worst terminal or canonical singularities in dimension four. As a corollary we also classify all four-dimensional one-point lattice simplices up to equivalence. Finally, we classify the terminal Gorenstein weighted projective spaces up to dimension ten.

Rational Points on Weighted projective Spaces

In this paper,we count the rational points on the weighted projective spaces defined over number fields w.r.t. ``size''. An asymptotic formula which generalizes the result of Schanuel's ``Heights in number fields'' is obtained. Furthermore, we count also the rational points on the product of weighted projective spaces.

Quantum cohomology of smooth complete intersections in weighted projective spaces and singular toric varieties

We generalize Givental's Theorem for complete intersections in smooth toric varieties in the Fano case. In particular, we find Gromov--Witten invariants of Fano varieties of dimension $\geq 3$, which are complete intersections in weighted projective spaces and singular toric varieties. We generalize the Riemann-Roch equations to weighted projective spaces. We compute counting matrices of smooth Fano threefolds with Picard group Z and anticanonical degrees 2, 8, and 16.

Mutations of Fake Weighted Projective Spaces

We characterise mutations between fake weighted projective spaces, and give explicit formulas for how the weights and multiplicity change under mutation. In particular, we prove that multiplicity-preserving mutations between fake weighted projective spaces are mutations over edges of the corresponding simplices. As an application, we analyse the canonical and terminal fake weighted projective spaces of maximal degree.

关于伪加权射影3-空间中曲面的一般Picard数的一个结果

Certain degenerations of surfaces in fake weighted projective $3$-space

本文给出了伪加权射影3-空间中某些一般非退化曲面具有Picard数大于1的判别准则，这些代数曲面为一般型，通过考虑沿边的退化并跟踪几何亏格，利用消失上同调类构造了与典范除子不成比例的有理Picard类。

Weighted Projective Spaces from the toric point of view with computational applications

The purpose of the present paper is threefold. First: giving a treatise on weighted projective spaces by the toric point of view. Second: providing characterizations of fans and polytopes giving weighted projective spaces, with particular focus on a kind of \emph{recognition process} of toric data like fans and polytopes. Third: building a mathematical framework for the algorithmic and computational approach to wps's realized in [23,24].

Hypergeometric Equations and Weighted Projective Spaces

We compute the Hodge numbers of the variation of Hodge structure of the middle cohomology (with compact support) of the Landau-Ginzburg model dual to a weighted projective space. We state a conjectural formula for the Hodge numbers of general hypergeometric variations. We show that a general Landau-Ginzburg fibre is birational to a Calabi-Yau variety if and only if a general anticanonical section of the weighted projective space is Calabi-Yau. We analyse the 104 weighted 3-spaces with canonical singularities, and show that a general anticanonical section is not a K3 surface exactly in those 9 …