Najmuddin Fakhruddin
Let X be a smooth projective toric surface and L and M two line bundles on X. If L is ample and M is generated by global sections, then we show that the natural map from H^0(X,L) tensor H^0(X,M) to H^0(X, L tensor M) is surjective. We also consider a generalization to the case when M is an arbitrary line bundle with h^0(X,M) > 0.
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We use multiplication maps to give a characteristic-free approach to vanishing theorems on toric varieties. Our approach is very elementary but is enough powerful to prove vanishing theorems.
We show that the moduli space of rank three toric vector bundles on smooth projective toric varieties satisfies Murphy's law, that is, every singularity type of finite type arises somewhere on this space.
We define a quasi--projective reduction of a complex algebraic variety $X$ to be a regular map from $X$ to a quasi--projective variety that is universal with respect to regular maps from $X$ to quasi--projective varieties. A toric quasi--projective reduction is the analogous notion in the category of toric varieties. For a given toric variety $X$ we first construct a toric quasi--projective reduction. Then we show that $X$ has a quasi--projective reduction if and only if its toric quasi--projective reduction is surjective. We apply this result to characterize when the action of a subtorus on a…
In this paper we show that quotients of smooth projective toric varieties by $μ_p$ in positive characteristics $p$ are toric varieties.
Using toric geometry we prove a Bézout type theorem for weighted projective spaces.
We give several mild conditions on a toric bundle on a nonsingular toric variety under which the projectivization of the toric bundle is Frobenius split.
On non-projective complete toric varieties
本文证明每个Picard数至多为五的光滑非射影完全环面三重簇在经历有限次flop或anti-flip后可变为射影簇。
For any $n\geq 3$, we explicitly construct smooth projective toric $n$-folds of Picard number $\geq 5$, where any nontrivial nef line bundles are big.
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}.
The automorphism group of a projective bundle P(E) over a simplicial toric variety is described when the bundle E is a direct sum of line bundles. Applications to study of moduli of complete intersections on toric varieties, including Calabi Yau m-folds and Del Pezzo surfaces, are indicated.
This is an expository paper in which we define projective GIT quotients and introduce toric varieties from this perspective. It is intended primarily for readers who are learning either invariant theory or toric geometry for the first time.
Motivated by the problem of classifying toric $2$-Fano manifolds, we introduce a new invariant for smooth projective toric varieties, the minimal projective bundle dimension. This invariant $m(X)\in\{1, \dots,\dim(X)\}$ captures the minimal degree of a dominating family of rational curves on $X$ or, equivalently, the minimal length of a centrally symmetric primitive relation for the fan of $X$. We classify smooth projective toric varieties with $m(X)\geq \dim(X)-2$, and show that projective spaces are the only $2$-Fano manifolds among smooth projective toric varieties with $m(X)\in\{1, \dim(X)…
We provide a generalization of the algorithm of Eklund-Jost-Peterson for computing Segre classes of closed subschemes of projective k-space. The algorithm is here generalized to computing the Segre classes of closed subschemes of smooth projective toric varieties.
Firstly we show a generalization of the (1,1)-Lefschetz theorem for projective toric orbifolds and secondly we prove that on 2k-dimensional quasi-smooth hypersurfaces coming from quasi-smooth intersection surfaces, under the Cayley trick, every rational (k,k)-cohomology class is algebraic, i.e., the Hodge conjecture holds on them.
An affine local criterion for toric projective space bundles
该研究探讨等维环态射成为射影空间丛的条件，证明仿射刚性定理并将其应用于长极射线相关的环收缩，得到无需ℚ-因子性的射影空间丛定理。
Characteristic-free characterizations of projective space for smooth toric varieties
该研究证明光滑射影环簇切丛含正秩丰富局部自由子层时同构于射影空间，还建立了Beauville上同调刻画的环簇形式，且指出无环簇假设时相关刻画在特征2下不成立。
Let $L$ be an ample line bundle over a smooth projective toric surface $X$. Then $L$ corresponds to a very ample lattice polytope $P$ that encodes many geometric properties of $L$. In this article, by studying $P$, we will give some necessary and sufficient numerical criteria for the adjoint series $|K_X+L|$ to be either nef or (very) ample.
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.
We show that the isolated invariant branches globalize to algebraic curves, when we consider weak toric type complex hyperbolic foliations on projective toric ambient surfaces. To do it, we pass through a characterization of weak toric type foliations in terms of "non-degeneracy" conditions, associated to Newton polygons. We also give a description of the relationship between invariant algebraic curves and isolated invariant branches, valid for the case of toric type, by means of the following dichotomy. Either there is a rational first integral and there are no isolated invariant branches or …
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].
Algebraic hyperbolicity for surfaces in smooth projective toric threefolds with Picard rank 2 and 3
本文研究Picard秩为2和3的光滑射影环面三维体中曲面的代数双曲性，利用组合描述与Haase和Ilten的方法，在这些簇中确定了众多代数双曲性曲面。
Let $\mathbb{G}_a$ be the additive group of the field of complex numbers $\mathbb{C}$. We say that an irreducible algebraic variety $X$ of dimension $n$ admits an additive action if there is a regular action of the group $\mathbb{G}_a^n = \mathbb{G}_a \times \ldots \times \mathbb{G}_a$ ($n$ times) on $X$ with an open orbit. In 2017 Baohua Fu and Jun-Muk Hwang introduced a class of Euler-symmetric varieties. They gave a classification of Euler-symmetric varieties and proved that any Euler-symmetric variety admits an additive action. In this paper we show that in the case of projective toric var…
Let $\mathbb{K}$ be an algebraically closed field of characteristic zero and $\mathbb{G}_a$ be the additive group of $\mathbb{K}$. We say that an irreducible algebraic variety $X$ of dimension $n$ over the field $\mathbb{K}$ admits an additive action if there is a regular action of the group $\mathbb{G}_a^n = \mathbb{G}_a \times \ldots \times \mathbb{G}_a$ ($n$ times) on $X$ with an open orbit. In this paper we find all projective toric hypersurfaces admitting additive action.
The present paper is devoted to generalizing, inside the class of projective toric varieties, the classification [Batyrev91], performed by Batyrev in 1991 for smooth complete toric varieties, to the singular $Q$--factorial case. Moreover, in the first part of the paper the Kleinschmidt classification of smooth complete toric varieties of Picard number 2 [Kleinschmidt] is revised.
We totally classify the projective toric varieties whose canonical divisors are divisible by their dimensions. In Appendix, we show that Reid's toric Mori theory implies Mabuchi's characterization of the projective space for toric varieties.
Toric subvarieties of projective space are classified up to projective automorphisms.
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.
We show that the second Chern character of any projective toric manifold of Picard number three is not ample. In connection with this result, we give various examples of the positivity of higher Chern characters of projective toric manifolds.
We give a simple combinatorial proof of the toric version of Mori's theorem that the only $n$-dimensional smooth projective varieties with ample tangent bundle are the projective spaces $\mathbb{P}^n$.
The notion of higher order dual varieties of a projective variety, introduced in \cite{P83}, is a natural generalization of the classical notion of projective duality. In this paper we present geometric and combinatorial characterizations of those equivariant projective toric embeddings that satisfy higher order selfduality. We also give several examples and general constructions. In particular, we highlight the relation with Cayley-Bacharach questions and with Cayley configurations.
The space of torus translations and degenerations of a projective toric variety forms a toric variety associated to the secondary fan of the integer points in the polytope corresponding to the toric variety. This is used to identify a moduli space of real degenerations with the secondary polytope. A configuration A of real vectors gives an irrational projective toric variety in a simplex. We identify a space of translations and degenerations of the irrational projective toric variety with the secondary polytope of A. For this, we develop a theory of irrational toric varieties associated to arb…
Exceptional Collections for Toric Fano Fivefolds
研究toric Fano五重形在五维时，利用对角线解析是否产生线丛的例外集合，并首次计数出300个符合条件的五重形。
We will use toric degenerations of the projective plane ${\mathbb{P}^ 2}$ to give a new proof of the triple points interpolation problems in the projective plane. We also give a complete list of toric surfaces that are useful as components in this degeneration.
Given a smooth and projective curve C and a smooth and projective toric variety X, we first describe a compactification of the space of morphisms from C to X representing a fixed homology class, and after we study the intersection theory on this variety.
We give the full classification of smooth toric Legendrian subvarieties in projective space. We also prove that under some minor assumptions the group of linear automorphisms preserving given Legendrian subvariety preserves the contact structure of the ambient projective space.
In response to a question of Reid, we find all anti-canonical Calabi-Yau hypersurfaces $X$ in toric weighted projective bundles over the projective line where the general fiber is a weighted K3 hypersurface. This gives a direct generalization of Reid's discovery of the 95 families of weighted K3 hypersurfaces. We also treat the case where $X$ is fibered over the plane with general fiber a genus one curve in a weighted projective plane.
With the help of Mori theory for projective toric manifolds due to M. Reid, we study non projective toric manifolds which become projective after a single blow up along an invariant curve.
A general theory for constructing linear secret sharing schemes over a finite field $\Fq$ from toric varieties is introduced. The number of players can be as large as $(q-1)^r-1$ for $r\geq 1$. We present general methods for obtaining the reconstruction and privacy thresholds as well as conditions for multiplication on the associated secret sharing schemes. In particular we apply the method on certain toric surfaces. The main results are ideal linear secret sharing schemes where the number of players can be as large as $(q-1)^2-1$. We determine bounds for the reconstruction and privacy thresho…
Exceptional Collections for Toric Fano Fourfolds
本文推广了对角分解的相关结果，研究了四维光滑射影环面法诺簇的例外系，发现124个簇中有72个可通过 Hanlon-Hicks-Lazarev 对角分解得到完全强线丛例外系，且与 Bondal 数值判据等价。
Let X be a complete toric variety and let Y be a smooth projective variety with Picard number one. We prove that, if there exists a surjective morphism from X to Y, then Y is a projective space.
A general problem in complex cobordism theory is to find useful representatives for cobordism classes. One particularly convenient class of complex manifolds consists of smooth projective toric varieties. The bijective correspondence between these varieties and smooth polytopes allows us to examine which complex cobordism classes contain a smooth projective toric variety by studying the combinatorics of polytopes. These combinatorial properties determine obstructions to a complex cobordism class containing a smooth projective toric variety. However, the obstructions are only necessary conditio…
In this survey I summarize the constructions of toric degenerations obtained from valuations and Gröbner theory and describe in which sense they are equivalent. I show how adapted bases can be used to generalize the classical Newton polytope to what is called a $\mathbb B$-Newton polytope. The $\mathbb B$-Newton polytope determines the Newton--Okounkov polytopes of all Khovanskii-finite valuations sharing the adapted standard monomial basis $\mathbb B$.
Rational Curves in Projective Toric Varieties
本文研究了射影托利丘变种中有理曲线的构造与性质，提出度数为d的凯莱结构，并建立了希尔伯特方案与最大光滑凯莱结构之间的对应关系。
Irreducible components of moduli spaces of maps to smooth projective toric varieties in genus 0
本文通过阿贝尔锥的不可约分支研究，给出了亏格0到光滑射影环簇的映射模空间不可约分支的组合描述，并首次描述了非射影空间情形的可光滑轨迹。
Let $P_m(J)$ denote a simplicial complex obtainable from consecutive wedge operations from an $m$-gon. In this paper, we completely classify toric manifolds over $P_m(J)$ and prove that all of them are projective. As a consequence, we provide an infinite family of projective toric manifolds.
A remark on toric foliations
本文研究射影 Q-因子环面簇上的环面叶状结构，证明当极射线长度超过叶状结构秩时，极收缩为射影空间丛且叶状结构等于其相对切层。
Let T be a torus over an algebraically closed field k of characteristic 0, and consider a projective T-module P(V). We determine when a projective toric subvariety X of P(V) is self-dual, in terms of the configuration of weights of V.
In this paper we illustrate an algorithmic procedure which allows to build projective wonderful models for the complement of a toric arrangement in a n-dimensional algebraic torus T. The main step of the construction is a combinatorial algorithm that produces a toric variety by subdividing in a suitable way a given smooth fan.
On the instability of syzygy bundles on toric surfaces
本文研究环面上合冲束的斜率稳定性，证明除两类特殊环面外，总可选取极化使高次幂线丛对应的合冲束失稳。
In this paper, using Klyachko's classification theorem we study positivity and semi-stability of toric vector bundles on a class of nonsingular projective toric varieties, known as Bott towers. In particular, we give a criterion of $s$-jet ampleness of line bundles and characterize nef and big line bundles on Bott towers using Bott numbers. We obtain a criterion for the ampleness of discriminant zero semi-stable toric vector bundles on nonsingular projective varieties. We also describe toric subbundles of toric vector bundles on nonsingular toric varieties.
Ample line bundles on certain toric fibered 3-folds
本文研究具有到射影直线的满射环面等变态射的射影非奇异环面3维流形，证明了其上的丰富线丛总是正规生成。
We discuss an experimental approach to open problems in toric geometry: are smooth projective toric varieties (i) projectively normal and (ii) defined by degree 2 equations? We discuss the creation of lattice polytopes defining smooth toric varieties as well as algorithms checking properties (i) and (ii) and further potential properties, in Particular a weaker version of (ii) asking for scheme-theoretic definition in degree 2.
The Cox ring provides a coordinate system on a toric variety analogous to the homogeneous coordinate ring of projective space. Rational maps between projective spaces are described using polynomials in the coordinate ring, and we generalise this to toric varieties, providing a unified description of arbitrary rational maps between toric arieties in terms of their Cox coordinates. Introducing formal roots of polynomials is necessary even in the simplest examples.
In this paper we show that a smooth toric variety $X$ of Picard number $r\leq 3$ always admits a nef primitive collection supported on a hyperplane admitting non-trivial intersection with the cone $\Nef(X)$ of numerically effective divisors and cutting a facet of the pseudo-effective cone $\Eff(X)$, that is $\Nef(X)\cap\partial\overline{\Eff}(X)\neq\{0\}$. In particular this means that $X$ admits non-trivial and non-big numerically effective divisors. Geometrically this guarantees the existence of a fiber type contraction morphism over a smooth toric variety of dimension and Picard number lowe…
Toric structure of the moduli space of points in projective space
研究通过构造射影空间中点模空间的托尔紧化，揭示了其扇形结构与permutohedron的关系，并推广了Losev-Manin空间与permutohedron的联系到任意维度。
We produce a decomposition of the parameter space of the $A$-hypergeometric system associated to a projective monomial curve as a union of an arrangement of lines and its complement, in such a way that the analytic behavior of the solutions of the system is explicitly controlled within each term of the union.
We show that the moduli space of stable n-pointed rational curves can be flatly degenerated to a projective toric variety. We arrive at this by showing that the Chow quotients of the Grassmannians admit toric degenerations, which in turn, follows from a theorem that we prove for toric degenerations of more general Chow quotients. Along the way, we also argued that GIT quotient of a flat family is again flat. In particular, all GIT quotients of flag varieties by maximal tori can be flatly degenerated to projective toric varieties ([9] and [12]).
This paper is devoted to settle two still open problems, connected with the existence of ample and nef divisors on a Q-factorial complete toric variety. The first problem is about the existence of ample divisors when the Picard number is 2: we give a positive answer to this question, by studying the secondary fan by means of Z-linear Gale duality. The second problem is about the minimum value of the Picard number allowing the vanishing of the Nef cone: we present a 3-dimensional example showing that this value cannot be greater then 3, which, under the previous result, is also the minimum valu…
We prove the Kawamata-Viehweg vanishing and another Kodaira-type vanishing for projective toric surfaces over arbitrary fields.
We show that very general hypersurfaces in odd-dimensional simplicial projective toric varieties verifying a certain combinatorial property satisfy the Hodge conjecture (these include projective spaces). This gives a connection between the Oda conjecture and Hodge conjecture. We also give an explicit criterion which depends on the degree for very general hypersurfaces for the combinatorial condition to be verified.
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