Boris Pasquier
We classify all smooth projective horospherical varieties with Picard number 1. We prove that the automorphism group of any such variety X acts with at most two orbits and that this group still acts with only two orbits on X blown up at the closed orbit. We characterize all smooth projective two-orbits varieties with Picard number 1 that satisfy this latter property.
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The aim of this article is the construction of the interior motive of a Picard variety. Those are Shimura varieties of PEL type. Our result is an application of the strategy developed by Wildeshaus to construct a Hecke-invariant motive whose realizations correspond to interior cohomology.
We prove the birational boundedness of Q-Fano varieties with Picard number one and dimension $n$ for every $n$. We do not use Mori theory.
Stability of syzygy bundles on varieties of Picard number one
本文利用Hilbert多项式给出了Picard数为1的光滑射影簇上整体生成丰沛线丛的合取丛斜率稳定性判据，并证明了其在Fano、Calabi--Yau及超凯勒簇等多种簇上的稳定性。
We describe smooth projective horospherical varieties with Picard number 1. Moreover we prove that the automorphism group of any such variety acts with at most two orbits and we give a geometric characterisation of non-homogeneous ones.
Symmetric varieties are normal equivariant open embeddings of symmetric homogeneous spaces and they are interesting examples of spherical varieties. The principal goal of this article is to study the rigidity under Kähler deformations of smooth projective symmetric varieties with Picard number one.
In this paper, we study derived categories of certain toric varieties with Picard number three that are blowing-up another toric varieties along their torus invariant loci of codimension at most three. We construct strong full exceptional collections by using Orlov's blow-up formula and mutations.
In this paper we study the geometry of mildly singular Fano varieties on which there is an effective prime divisor of Picard number one. Afterwards, we address the case of toric varieties. Finally, we treat the lifting of extremal contractions to universal covering spaces in codimension 1.
We classify all smooth projective horospherical varieties of Picard group $\mathbb{Z}^2$ and we give a first description of their geometry via the Log Minimal Model Program.
In this paper, we study the set $R_g^{(p)}$ of possible Picard numbers of abelian varieties of dimension $g$ over algebraically closed fields of characteristic $p>0$. We show that many of the results for complex abelian varieties have analogues in positive characteristic: non-completeness in dimension $g \geq 2$, asymptotic completeness as $g \rightarrow +\infty$, structure results for abelian varieties of large Picard number. On the way, we highlight and discuss new characteristic $p>0$ features and pathologies: non-additivity of the range of Picard numbers, supersingularity index of an abeli…
We investigate full strongly exceptional collections on smooth, com- plete toric varieties. We obtain explicit results for a large family of varieties with Picard number three, containing many of the families already known. We also describe the relations between the collections and the split of the push forward of the trivial line bundle by the toric Frobenius morphism.
We compute the Picard group of the universal abelian variety over the moduli stack $\mathscr A_{g,n}$ of principally polarized abelian varieties over $\mathbb{C}$ with a symplectic principal level $n$-structure. We then prove that over $\mathbb{C}$ the statement of the Franchetta conjecture holds in a suitable form for $\mathscr A_{g,n}$.
We show that every connected commutative algebraic group over an algebraically closed field of characteristic 0 is the Picard variety of some projective variety, obtained by pinching a smooth variety along a finite subscheme. In contrast, no Witt group of dimension at least 3 over a perfect field of prime characteristic is isogenous to the Picard variety obtained by this construction.
We study the birationality (onto its image) of the Abel-Prym morphism associated with a Prym-Tuyrin variety. We use such result to prove that Picard bundles over Prym varieties are simple and moreover they are stable when the Abel-Prym morphism is not birational. As a consequence we obtain that Picard bundles over moduli spaces of stable vector bundles with fixed determinant are simple. We prove that Picard bundles over moduli spaces of rank 2 vector bundles on curves of genus 2 bundles are stable.
We study the possible Picard numbers of abelian varieties of given dimension $g$. If $R_g$ denotes the set of realizable Picard numbers, then $R_g$ is bounded by $g^2$. We show that, for $g$ at least $3$, the set $R_g$ always has gaps and we analyze the nature of these gaps. We further prove that the Picard numbers are asymptotically complete in $[1,g^2]$ as $g$ goes to infinity. Finally we show that every Picard number which can be realized over the complex numbers can already be realized by an abelian variety defined over a number field.
Recently, Kanemitsu has discovered a counterexample to the long-standing conjecture that the tangent bundle of a Fano manifold of Picard number one is (semi)stable. His counterexample is a smooth horospherical variety. There is a weaker conjecture that the tangent bundle of a Fano manifold of Picard number one is simple. We prove that this weaker conjecture is valid for smooth horospherical varieties of Picard number one. Our proof follows from the existence of an irreducible family of unbendable rational curves whose tangent vectors span the tangent spaces of the horospherical variety at gene…
Smooth projective varieties with a torus action of complexity 1 and Picard number 2
本文显式描述了具有复杂度为1的环面作用且皮卡德数至多为2的光滑簇，并分类了各维度皮卡德数为2的光滑（几乎）法诺簇，证明其均可由维数至多7的光滑簇经迭代广义锥构造得到。
We prove that the Picard group of a regular simply connected variety over an algebraically closed field of arbitrary characteristic is finitely generated. The main difficulty to overcome is the unavailability of resolution of singularities. From this we deduce that in positive characteristic there exist no nontrivial stratified line bundles on such a variety, and we present a complex analog.
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].
Correspondences for hyperkähler varieties with large Picard numbers
研究具有大皮卡数的超凯勒流形与阿贝尔簇的联系，核心方法是推广莫里森对（修正）小田猜想的解答，主要贡献是将该定理推广到K3曲面上的点希尔伯特概型等超凯勒流形例子。
We construct a full, strongly exceptional collection of line bundles on the variety X that is the blow up of the projectivization of the vector bundle O_{P^{n-1}}\oplus O_{P^{n-1}}(b) along a linear space of dimension n-2, where b is a non-negative integer.
We give a complete classification of P1-bundles over a projective manifold of Picard number one which admit another smooth morphism of relative dimension one.
In this paper, we give the complete classification of full exceptional collections on smooth toric Fano threefolds and fourfolds with Picard rank two. To be precise, we give a partial answer to the conjecture in \cite{Kuz} and \cite{LYY}: we provide all the exceptional collections of maximal length and prove that they are in fact full exceptional collections.
Classifying weak Fano toric varieties of Picard rank $3$
本文提出一种使用 Macaulay2 系统分类任意维数下 Picard 秩为 $3$ 的光滑弱 Fano 环面簇的方法，并显式给出了维数 $3$ 和 $4$ 的分类结果。
Complete non-singular toric varieties with Picard number 4
本文用扇形单纯球和特征映射组合框架分类了 Picard 数为四的完备非奇异环面簇，得到 59 个种子，解决相关猜想并回答 Batyrev 的问题。
We give a combinatorial criterion for the tangent bundle on a smooth toric variety to be stable with respect to a given polarisation in terms of the corresponding lattice polytope. Furthermore, we show that for a smooth toric surface and a smooth toric variety of Picard rank 2, there exists an ample line bundle with respect to which the tangent bundle is stable if and only if it is an iterated blow-up of projective space.
Projective homogeneous varieties of Picard rank one in small characteristic
本文将 Picard 秩为 1 的射影齐次簇分类推广至特征 2 和 3，给出 $G_2$ 型特征 2 下的唯一反例以完成分类，并构造了 Picard 秩为 2 的新实例。
Over a family of varieties with singular special fiber, the relative Picard functor (i.e. the moduli space of line bundles) may fail to be compact. We propose a stability condition for line bundles on reducible varieties that is aimed at compactifying it. This stability condition generalizes the notion of `balanced multidegree' used by Caporaso in compactifying the relative Picard functor over families of curves. Unlike the latter, it is defined `asymptotically'; an important theme of this paper is that although line bundles on higher-dimensional varieties are more complicated than those on cu…
We generalize the Harnack-Thom theorem to relate the ranks of the Lawson homology groups with $\Z_2$-coefficients of a real quasiprojective variety with the ranks of its reduced real Lawson homology groups. In the case of zero-cycle group, we recover the classical Harnack-Thom theorem and generalize the classical version to include real quasiprojective varieties. We use Weil's construction of Picard varieties to construct reduced real Picard groups, and Milnor's construction of universal bundles to construct some weak models of classifying spaces of some cycle groups. These weak models are use…
Picard Groups of Spectral Varieties and Moduli of Higgs Sheaves
该研究针对曲面上的希格斯层模空间与希钦映射，证明了高维谱簇的Noether–Lefschetz型定理，给出一般希钦纤维非空的充要条件，并分析第二陈类变化对模空间几何的影响。
In this paper, we study arithmetically Cohen--Macaulay (ACM) bundles on homogeneous varieties $G/P$. Indeed we characterize the homogeneous ACM bundles on $G/P$ of Picard rank one in terms of highest weights. This is a generalization of the result on $G/P$ of classical types presented by Costa and Miró-Roig for type $A$, and Du, Fang, and Ren for types $B,C$ and $D$. As a consequence we prove that only finitely many irreducible homogeneous ACM bundles, up to twisting line bundles, exist over all such $G/P$. Moreover, we derive the list of the highest weights of the irreducible homogeneous ACM …
We study the Picard rank of smooth toric Fano varieties possessing families of minimal rational curves of given degree. We discuss variants of a conjecture of Chen-Fu-Hwang and prove a version of their statement that recovers the original conjecture in sufficiently high dimension. We also prove new cases of the original conjecture for high degrees in all dimensions. Our main tools come from toric Mori theory and the combinatorics of Fano polytopes.
We study the Zariski closure of the monodromy group $\mathbf{Mon}$ of Lauricella's hypergeometric function $F_C$. If the identity component $\mathbf{Mon}^0$ acts irreducibly, then $\overline{\mathbf{Mon}} \cap \mathbf{SL}_{2^n}(\mathbb{C})$ must be one of classical groups $\mathbf{SL}_{2^n}(\mathbb{C}), \mathbf{SO}_{2^n}(\mathbb{C})$ and $\mathbf{Sp}_{2^n}(\mathbb{C})$. We also study Calabi-Yau varieties arising from integral representations of $F_C$.
In this paper we study the geometry of the $14$ families of K3 surfaces of Picard number four with finite automorphism group, whose Néron-Severi lattices have been classified by È.B. Vinberg. We provide projective models, we identify the degrees of a generating set of the Cox ring and in some cases we prove the unirationality of the associated moduli space.
We provide a criterion for when Hilbert squares of complex projective K3 surfaces with Picard number one are strongly ambiguous. This criterion is the same as [DM, Proposition 3.14], but is obtained by a different method. In particular, this enables us to compute the automorphism groups of these Hilbert squares from a different point of view with respect to [BCNS].
The special Brauer group and twisted Picard varieties
本文通过引入Grothendieck的特殊Brauer群，将带截面椭圆K3曲面的Tate-Shafarevich群概念推广至赋予线性系统的K3曲面，为处理扭曲层模空间提供了有效方法。
Property $\mathcal{O}$ for an arbitrary complex, Fano manifold $X$, is a statement about the eigenvalues of the linear operator obtained from the quantum multiplication of the anticanonical class of $X$. Conjecture $\mathcal{O}$ is a conjecture that Property $\mathcal{O}$ holds for any Fano variety. Pasquier listed the smooth non-homogeneous horospherical varieties of Picard rank 1 into five classes. Conjecture $\mathcal{O}$ has already been shown to hold for the odd symplectic Grassmannians which is one of these classes. We will show that Conjecture $\mathcal{O}$ holds for two more classes an…
Stability of the canonical extension of tangent bundles on Picard-rank-1 Fano varieties
本文研究Picard数为1的Fano簇上切丛典范扩张的斜率稳定性，证明在指数整除维数或维数加一时切丛稳定性蕴含典范扩张的（半）稳定性，并给出其在曲线稳定向量丛模空间上的应用推论。
Let $X$ be a $\Q$-factorial complete toric variety over an algebraic closed field of characteristic $0$. There is a canonical injection of the Picard group ${\rm Pic}(X)$ in the group ${\rm Cl}(X)$ of classes of Weil divisors. These two groups are finitely generated abelian groups; whilst the first one is a free group, the second one may have torsion. We investigate algebraic and geometrical conditions under which the image of ${\rm Pic}(X)$ in ${\rm Cl}(X)$ is contained in a free part of the latter group.
Let X be a Q-factorial Gorenstein Fano variety. Suppose that the singularities of X are canonical and that the locus where they are non-terminal has dimension zero. Let D be a prime divisor of X. We show that rho_X - rho_D < 9 (where rho is the Picard number). Moreover, if rho_X - rho_D > 3, there exists a finite morphism from X to S x Y, where S is a surface with rho_S at most 9. As an application we prove that, if X has dimension 3, then rho_X is at most 10.
On Primitive Ulrich Bundles over a few projective varieties with Picard number two
本文在光滑射影簇中引入本原 Ulrich 丛的概念，并在次数 $6$ 的旗三维簇和有理法向滚动面情形下给出了其上同调刻画，同时提出了若干开放问题。
We calculate the Picard group of generic (very general) spectral varieties living in the total space of a very ample line bundle over an algebraically closed field $k$ of odd characteristics or characteristic 0. We follow the strategy of Ravindra and Srinivas [RS06,RS09] via formal Picard groups. As an application, we calculate the generic fibers of Hitchin systems over a smooth quintic surface of Picard number 1.
Smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 are horospherical varieties. We characterize standard embeddings of smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 by means of varieties of minimal rational tangents. In particular, we mainly consider nonhomogeneous smooth Schubert varieties in symplectic Grassmannians and in the 20-dimensional $F_4$-homogeneous manifold associated to a short simple root.
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。
Stable and unstable syzygy bundles on certain Picard rank one varieties
研究维数至少为\(2\)且皮卡秩为\(1\)的光滑射影簇上合冲丛稳定性，证明在此类簇上丰富线丛合冲丛稳定，尤其适用于射影空间完全交，扩展了前人成果。
We study the structure of rational Picard groups of hypersurfaces of toric varieties. By using the fan structure associated to the ambient toric variety, an explicit basis of the Picard group is described by certain combinatorial data. We shall also discuss the application to Calabi-Yau spaces.
Let S be a smooth, projective surface of Picard rank 1 and very ample generator embedding S into P^n. Let C be a smooth curve in O(m) for m \geq 5. We prove that any base-point free, complete g^r_d on C for r\in\{1,2\} and d small enough is cut out by a hyperplane section restricted to a multisecant (n-r-1)-plane.
We give sufficient conditions for the semisimplicity of quantum cohomology of Fano varieties of Picard rank 1. We apply these techniques to prove new semisimplicity results for some Fano varieties of Picard rank 1 and large index. We also give examples of Fano varieties having a non semisimple small quantum cohomology but a semisimple big quantum cohomology.
The following conjecture arose out of discussions between B. Harbourne, J. Roé, C. Cilberto and R. Miranda: for a smooth projective surface $X$ there exists a positive constant $c_X$ such that $h^1(\mathcal O_X(C))\le c_X h^0(\mathcal O_X(C))$ for every prime divisor $C$ on $X$. When the Picard number $ρ(X)=2$, we prove that if either the Kodaira dimension $κ(X)=1$ and $X$ has a negative curve or $X$ has two negative curves, then this conjecture holds for $X$.
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…
Stability of Equivariant Logarithmic Tangent Sheaves on Toric Varieties of Picard Rank Two
本文研究了正规环面簇上等变对数切层的斜率稳定性，并完整刻画了Picard秩为一或二时除子与极化使得该层（半）稳定的条件。
Homogeneous Ulrich bundles on homogenous varieties of certain exceptional types
本文针对有理齐性簇上的齐性向量丛，提出了适用于任意Picard数的Ulrich性判别准则，推广了Fonarev的Picard数1情形，并证明某些例外型齐性簇在最小丰沛类下不存在齐性Ulrich丛。
Let $U^{'s}_L(n,d)$ be the moduli space of stable vector bundles of rank $n$ with determinant $L$ where $L$ is a fixed line bundle of degree $d$ over a nodal curve $Y$. We prove that the projective Poincare bundle on $Y \times U^{'s}_L(n,d)$ and the projective Picard bundle on $U^{'s}_L(n,d)$ are stable for suitable polarisation. For a nonsingular point $x \in Y$, we show that the restriction of the projective Poincare bundle to $x \times U^{'s}_L(n,d)$ is stable for any polarisation. We prove that for arithmetic genus $g\ge 3$ and for $g=n=2, d$ odd, the Picard group of the moduli space $U'_L…
On Chow and Picard groups of Richardson varieties
本文研究投影Richardson簇的Chow与Picard群，证明其Pic群到1次周群对偶的次数映射为单射，在多数特征下为同构，猜想所有特征下均成立，采用几何方法完成证明。
Line-filtered rank-two ACM bundles on abelian varieties of Picard rank one
针对皮卡秩1的维数≥2的复阿贝尔簇，分类其上空亏格的秩2算术Cohen-Macaulay丛，明确其结构形式并证明这些丛均非Ulrich丛。
Geometric structures modeled on rational homogeneous manifolds are studied to characterize rational homogeneous manifolds and to prove their deformation rigidity. To generalize these characterizations and deformation rigidity results to quasihomogeneous varieties, we first study horospherical varieties and geometric structures modeled on horospherical varieties. Using Cartan geometry, we prove that a geometric structure modeled on a smooth projective horospherical variety of Picard number one is locally equivalent to the standard geometric structure when the geometric structure is defined on a…
This note is a report on the observation that some singular varieties admit Calabi--Yau coverings. As an application, we construct 18 new Calabi--Yau 3-folds with Picard number one that have some interesting properties.
It is conjectured that a Fano manifold of Picard number 1 which is not a projective space admits no endomorphisms of degree bigger than 1. Beauville confirmed this for hypersurfaces of projective space. We study this problem for hypersurfaces of an arbitrary Fano manifold of Picard number 1.
$c_1$-cohomological rigidity for smooth toric Fano varieties of Picard number two
本研究针对Picard数为2的光滑环面Fano簇，验证了$c_1$-上同调刚性猜想，即若两簇的整上同调环有保$c_1$同构则簇同构，推进了代数几何领域相关刚性问题的研究。
Local rigidity of projective smooth horospherical varieties of Picard number two
研究 Picard 数为二的射影光滑 horospherical 变种的局部刚性，基于先前工作中的自同构群计算结果。
In a previous paper we have classified the smooth projective symmetric G-varieties with Picard number one (and G semisimple). In this work we give a geometrical description of such varieties. In particular, we determine their group of automorphisms. When this group, Aut(X), acts non-transitively on X, we describe a G-equivariant embedding of the variety X in a homogeneous variety (with respect to a larger group).
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