Jiun-Cheng Chen, Hsian-Hua Tseng
We prove a numerical characterization of $\mathbb{P}^n$ for varieties with at worst isolated local complete intersection quotient singularities. In dimension three, we prove such a numerical characterization of $\mathbb{P}^3$ for normal $\mathbb{Q}$-Gorenstein projective varieties.
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We give some results on quadratic normality of reducible curves canonically embedded and partially extend this study to their projective normality.
A note on the converse Bricard property of projective planes
本文通过构造反例证明逆Bricard性质在一般Moufang平面中不成立，澄清了该性质与Moufang平面之间的关系。
We characterize the integral Zariski decomposition of a smooth projective surface with Picard number 2 to partially solve a problem of B. Harbourne, P. Pokora, and H. Tutaj-Gasinska [Electron. Res. Announc. Math. Sci. 22 (2015), 103--108].
A note on stability conditions on projective spaces
给出李定理关于射影空间上凝聚层有界导出范畴几何布里奇兰德稳定性条件存在性的新证明，此条件可限制到任意光滑射影簇诱导布里奇兰德稳定性条件。
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.
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.
Notes on cylinders in smooth projective surfaces
研究完美域上光滑极小几何有理曲面中柱面的存在条件，并证明柱面的存在性在光滑射影曲面的双有理映射下保持不变。
We prove a formula for the orbifold Chow ring of semi-projective toric DM stacks, generalizing the orbifold Chow ring formula of projective toric DM stacks by Borisov-Chen-Smith. We also consider a special kind of semi-projective toric DM stacks, the Lawrence toric DM stacks. We prove that the orbifold Chow ring of a Lawrence toric DM stack is isomorphic to the orbifold Chow ring of its associated hypertoric DM stack studied in \cite{JT}.
Being inspired by a work of Curtis T. McMullen about a very impressive automorphism of a K3 surface of Picard number zero, we shall clarify the structure of the bimeromorphic automorphism group of a non-projective hyperkähler manifold, up to finite group factor. We also discuss relevant topics, especially, new counterexamples of Kodaira's problem about algebraic approximation of a compact Kähler manifold.
We determine the second fundamental form of a variation of Hodge Structure of a smooth projective hypersurface using the classical identification of the Hodge structure and the action of the infinitesimal variation of Hodge structure with pieces of the Jacobian Ideal of the hypersurface. This simple algebraic description also implies that, in this case, the variation of Hodge structure satisfies a new set of second order partial differential equations.Comments are welcome!
It is well known that for an ample line bundle $L$ on an Abelian variety $A$, the linear system |2L| is base point free, and 3L is very ample, moreover the map defined by the linear system |2L| is well understood. In this paper we generalized this classical result and give a new proof using the theory CGG developed by Pareschi and Popa.
A note on the primitive cohomology lattice of a projective surface
本文研究射影曲面的本原上同调格，证明在不定情形下它可由相交格和本原丰沛类的自交数确定，并给出 Godeaux 曲面、Kunev 曲面及 Horikawa 曲面等例子，同时讨论了一般型典范极化曲面的负定本原格结果。
We collect some results on endomorphisms on projective varieties related with the Kawaguchi-Silverman conjecture. We discuss certain condition on automorphism groups of projective varieties and positivity conditions on leading real eigendivisors of self-morphisms. We prove Kawaguchi-Silverman conjecture for endomorphisms on projective bundles on a smooth Fano variety of Picard number one. In the last section, we discuss endomorphisms and augmented base loci of their eigendivisors.
The pourpose of these notes is to give an elementary description of one of the simplest Berkovich's spaces: that associated to the projective line of a complete and algebraically closed ultrametric field. \ L'objectif de ces notes est de donner une description élémentaire de l'un des espaces de Berkovich le plus simple : celui associé à la droite projective, où le corps ultramétrique de base sera supposé complet et algébriquement clos.
A $\mathbf{Q}$-Cartier divisor $D$ on a projective variety $M$ is {\it almost nup}, if $(D , C) > 0$ for every very general curve $C$ on $M$. An algebraic variety $X$ is of {\it almost general type}, if there exists a projective variety $M$ with only terminal singularities such that the canonical divisor $K_M$ is almost nup and such that $M$ is birationally equivalent to $X$. We prove that a complex algebraic variety is of almost general type if and only if it is neither uniruled nor covered by any family of varieties being birationally equivalent to minimal varieties with numerically trivial …
Let ${\bf P}^n$ be the projective $n-$space over the complex numbers. In this note we show that an indecomposable rigid coherent sheaf on ${\bf P}^n$ has a trivial endomorphism algebra. This generalizes a result of Drezet for $n=2.$
These are notes on some algebraic geometry of complex projective curves, together with an application to studying the contact curves in CP^3 and the null curves in the complex quadric Q^3 in CP^4, related by the well-known Klein correspondence. Most of this note consists of recounting the classical background. The main application is the explicit classification of rational null curves of low degree in Q^3. I have recently received a number of requests for these notes, so I am posting them to make them generally available.
The aim of this note is to investigate the relation between two types of non-singular projective varieties of Picard rank 2, namely the Projective bundles over Projective spaces and certain Blow-up of Projective spaces.
In this note, we prove that every fibre space structures of a projective irreducible symplectic manifold is a lagrangian fibration.
Projective modules are a link between geometry and algebra as established by the theorem of Serre-Swan. In this paper, we define the super analog of projective modules and explore this link in the case of some particular super geometric objects. We consider the tangent bundle over the supersphere and show that the module of vector field over a supersphere is a super projective module over the ring of supersmooth functions. Also, we discuss a class of super projective modules that can be constructed from a projection map on modules defined over the ring of supersmooth functions over supersphere…
It has been shown by Hacking and Prokhorov that if the projective surface X with quotient singularities and self-intersection number 9 has a smoothing to the projective plane, then X is the general fiber of a Q-Gorenstein deformation of the weighted projective plane with weights giving solutions to the Markov equation. This result has been understood and generalized by combinatorial mutations of Fano triangles by Akhtar, Coates, Galkin, and Kasprzyk. In this note, we study this result by utilizing polarized T-varieties and describe the associated deformation explicitly in terms of certain Mink…
Let $f:X\to Y$ be a fibration from a smooth projective 3-fold to a smooth projective curve, over an algebraically closed field $k$ of characteristic $p >5$. We prove that if the generic fiber $X_η$ has big canonical divisor $K_{X_η}$, then $$κ(X)\geκ(Y) + κ(X_η).$$
In this short note we announce explicit equations of a fake projective plane in its bicanonical embedding in $\mathbb C\mathbb P^9$.
In this short note, we observe that Theorem 3.1 in arXiv:1508.00682 for semiorthogonal indecomposability of the derived category of smooth DM stacks based on the canonical bundle can be extended to the case of projective varieties with Cohen-Macaulay singularities. As a consequence, all projective curves of positive arithmetic genus have weakly indecomposable bounded derived categories and indecomposable categories of perfect complexes. Here weak indecomposablility refers to the admissibility of components.
Let H be a closed subgroup of a linear algebraic group G defined over a field F. There is an equivalence of categories between the category of linear finite-dimensional representations of H and the category of finite rank G-homogeneous vectorbundles on G/H. In this short note we study this correspondence for the sheaves of principal parts (=jetbundles) of homogeneous linebundles on projective space in characteristic zero. We describe the representation corresponding to the principal parts of a homogeneous linebundle and as a corollary we obtain known formulas on the splitting type of the princ…
A note on a gap in the proof of the minimum distance for Projective Reed-Muller Codes
本研究针对S.Ghorpade与R.Ludhani近期指出的射影里德-穆勒码最小距离证明中的疏漏，阐明原始证明思路并完成填补，完善了该码的最小距离证明体系。
In this note, we give an overview of a new technique for studying Brill--Noether curves in projective space via degeneration. In particular, we give a roadmap to the proof of the Maximal Rank Conjecture.
Entropy of categorical dynamics is defined by Dmitrov-Haiden-Katzarkov-Kontsevich. Motivated by the fundamental theorem of the topological entropy due to Gromov-Yomdin, it is natural to ask an equality between the entropy and the spectral radius of induced morphisms on the numerical Grothendieck group. In this paper, we add two results on this equality: the lower bound in a general setting and the equality for orbifold projective lines.
In this small note we present a Tannakian proof of the theorem of Grothendieck-Harder on the classification of torsors under a reductive group on the projective line over a field.
We prove a non-integrability result concerning iterative derivations on projective line, where the iterative rule is given by a non-algebraic formal group.
In this Lecture Notes we present, in a sufficiently self contained way, our contributions and interests in the field of Minimal Model Theory. We study Fano-Mori spaces, both from the biregular and the birational point of view. For the former we recall and develop Kawamata's Base Point Free technique and some of Mori's deformation arguments. For the latter we lean on Sarkisov and #-Minimal Model Programs. In writing these notes we want to give our point of view on this area of research. We are not trying to give a treatment of the whole subject. These notes collect some topics we presented in t…
We study the a-numbers and p-ranks of Kummer covers of the projective line, and we give bounds for these numbers.
In this note, we show that the canonical orientation data on the quiver hearts are compatible under the autoequivalence $\_\otimesπ^*O(1)$, and hence glue to give an orientation data for the stack of coherent sheaves on local P2.
These are lecture notes on the rigidity of submanifolds of projective space "resembling" compact Hermitian symmetric spaces in their homogeneous embeddings. Recent results are surveyed, along with their classical predecessors. The notes include an introduction to moving frames in projective geometry, an exposition of the Hwang-Yamaguchi ridgidity theorem and a new variant of the Hwang-Yamaguchi theorem.
The main purpose of the note is to exclude the existence of certain submaximal curves in fake projective planes. This will lead to lower bounds on multipoint Seshadri constants of `fake' $\mathcal{O}(1)$ on fake projective planes.
A common criticism of infinity-categories in algebraic geometry is that they are an extremely technical subject, so abstract to be useless in everyday mathematics. The aim of this note is to show in a classical example that quite the converse is true: even a naive intuition of what an infinity-groupoid should be clarifies several aspects of the infinitesimal behaviour of the periods map of a projective manifold. In particular, the notion of Cartan homotopy turns out to be completely natural from this perspective, and so classical results such as Griffiths' expression for the differential of th…
In this note we describe the image of $\PP^2$ in $ Gr(2, \CC^{4})$ under a morphism given by a rank two vector bundle on $\PP^2$ with Chern classes $(2,2).$
A note on the semistability of singular projective hypersurfaces
本文利用次数、奇点最大重数与奇异轨迹维数，并结合 Benoist 的交维数下界及 Hilbert–Mumford 判据，给出奇异射影超曲面半稳定的充分条件。
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.
We prove that any smooth rational projective surface over the field of complex numbers has an open covering consisting of 3 subsets isomorphic to affine planes.
We study Gauss maps of order $k$, associated to a projective variety $X$ embedded in projective space via a line bundle $L.$ We show that if $X$ is a smooth, complete complex variety and $L$ is a $k$-jet spanned line bundle on $X$, with $k\geq 1,$ then the Gauss map of order $k$ has finite fibers, unless $X=\mathbb{P}^n$ is embedded by the Veronese embedding of order $k$. In the case where $X$ is a toric variety, we give a combinatorial description of the Gauss maps of order $k$, its image and the generic fibers.
In this note we exhibit the so-called Harbourne constants which capture and measure the Bounded Negativity on various birational models of an algebraic surface. We show an estimation for Harbourne constants for conic configurations on the complex projective plane with transversal intersection points.
The algebra of symmetric tensors on smooth projective varieties
本文讨论光滑复射影簇上对称张量代数H^0(X, Sym*TX)，通过实例计算并给出其Krull维数精确界，同时提出对具伪有效切丛的非不可直纹射影流形的猜想刻画。
Let X be a smooth projective variety of dimension n on which rational and homological equivalence coincide for algebraic p-cycles in the range 0\leq p\leq s. We show that the homologically trivial sector of rational Lawson homology L_pH_k(X,Q)_{hom} vanishes for 0\leq n-p\leq s+2. This is an analogue of a theorem of C. Peters in "dual dimensions". Together with Peters' theorem we get that the natural transformation L_pH_k(X,Q)--> H_k(X,Q) is injective for all p and k when X is a smooth projective variety of dimension 4 and Ch_0(X) =Z.
A couple of complex projective plane curves are said to make a Zariski pair if they have the same degree and the same type of singularities, but their embeddings in the projective plane are topologically different. In this paper, we present a method to construct certain type of Zariski pairs, and make infinite series of examples.
A note on varieties of non-negative Kodaira dimension with polarized self maps
本文证明非负柯西利亚维度且具有极化自映射的光滑射影变种必为阿贝尔变种的有限自由商。
In this note we show that if an abelian variety possesses a Galois embedding into some projective space, then it must be isogenous to the self product of an elliptic curve. We prove moreover that the self product of an elliptic curve always has infinitely many Galois embeddings.
Let $\Cal E$ be a very ample vector bundle of rank two on a smooth complex projective threefold $X$. An inequality about the third Segre class of $\Cal E$ is provided when $K_X+\det \Cal E$ is nef but not big, and when a suitable positive multiple of $K_X+\det \Cal E$ defines a morphism $X\to B$ with connected fibers onto a smooth projective curve $B$, where $K_X$ is the canonical bundle of $X$. As an application, the case where the genus of $B$ is positive and $\Cal E$ has a global section whose zero locus is a smooth hyperelliptic curve of genus $\geq 2$ is investigated, and our previous res…
We calculate a Griffiths-type ring for smooth complete intersection in Grassmannians. This is the analogue of the classical Jacobian ring for complete intersections in projective space, and allows us to explicitly compute their Hodge groups.
We formulate and prove the existence of an asymptotic duality along the fibers of the Green-Griffiths jet bundles over projective manifolds. The existence of global sections for these bundles and also for their dual sheaves has been established by the author and J. P. Demailly in a previous article, [3].
We discuss the flatness property of some fiber type contractions of complex smooth projective varieties of arbitrary dimensions. We relate the flatness of some morphisms having one-dimensional fibers with their conic bundles structures, also in the general case in which some mild singularities of the varieties are admitted.
In this note we show that the multipoint Seshadri constant determines the maximum possible radii of embeddings of Kähler balls and vice versa.
Hartshorne's conjecture about vector bundles on projective space states that any rank 2 vector bundle on n-dimensional projective space splits as soon as n is at least 7. Klyachko has shown that Hartshorne's conjecture is true when the vector bundles are torus equivariant. Moreover, recent work of Ilten and Süss generalizes Klyachko's work to the case of a smaller rank torus action on projective space. In this note we give a new, direct proof that torus rank 2 bundles split that avoids a description of the category of torus equivariant vector bundles.
Algebraic magnetism invariants of a double scalar action on the projective plane
本文研究双标作用于射影平面的代数磁不变量，计算纯磁体和吸引子，并引入lambda可实现磁体的概念，同时解决了一些猜想。
Framed instanton pairs on the blow-up of the projective 3-space at a point
本文研究了射影3空间在一点处吹胀得到的Fano 3-fold上的Huybrechts和Lehn标架层，给出了3维中此类对的新例子，并证明了其模空间是fine、拟射影且无阻碍的。
In this short note, we prove the Miyaoka-Yau inequality for minimal projective $n$-manifolds of general type by using Kähler-Ricci flow.
Let $S$ be a smooth projective complex algebraic surface and $f\, :\, S\, \longrightarrow\, {\mathbb C}{\mathbb P}^2$ a finite map. Consider a pencil of hyperplane sections on ${\mathbb C}{\mathbb P}^2$ and pull it back to $S$. We address the question whether such a pencil is a Lefschetz pencil on $S$.
Let $X$ be a smooth projective connected curve of genus $g\ge 2$ defined over an algebraically closed field $k$ of characteristic $p>0$. Let $G$ be a finite group, $P$ a Sylow $p$-subgroup of $G$ and $N_G(P)$ its normalizer in $G$. We show that if there exists an étale Galois cover $Y\to X$ with group $N_G(P)$, then $G$ is the Galois group wan étale Galois cover $\mathcal{Y}\to\mathcal{X}$, where the genus of $\mathcal{X}$ depends on the order of $G$, the number of Sylow $p$-subgroups of $G$ and $g$. Suppose that $G$ is an extension of a group $H$ of order prime to $p$ by a $p$-group $P$ and $…
In this note we introduce the concept of reflective projective varieties. These are stratified projective varieties with certain dimension constraints on their dual varieties. We prove that for such varieties, the Chern-Schwartz-MacPherson classes of the strata completely determine the local Euler obstructions and the polar degrees. We also propose an algorithm to compute the local Euler obstructions when such varieties are formed by group orbits. As examples we compute the local Euler obstructions of quadratic hypersurfaces and ordinary determinantal varieties to illustrate our method.
A note on Zariski dense orbit conjecture
本文给出Zariski稠密轨道猜想的一个注记，简化了射影三维簇自同构的ZDO结果，并证明了不规则度足够大的射影簇的ZDO成立。
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