Miguel A. Barja, Francesco Zucconi
Let f:S ->B be a relatively minimal fibred surface. In this note we give a partial affirmative answer to a conjecture of Xiao, proving that the direct image of the relative dualizing sheaf of $f$ is ample when the slope of the fibration is less than 4, if the general fibre of $f$ is non-hyperelliptic or the genus of the fibre or of the base curve is low.
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A canonically fibered surface is a surface whose canonical series maps it to a curve. Using Miyaoka-Yau inequality, A. Beauville proved that a canonically fibered surface has relative genus at most 5 when its geometric genus is sufficiently large. G. Xiao further conjectured that the relative genus cannot exceed 4. We give a proof of this conjecture.
We prove that the genus $g$, the relative irregularity $q_f$ and the Clifford index $c_f$ of a non-isotrivial fibration $f$ satisfy the inequality $q_f \leq g - c_f$. This gives in particular a proof of Xiao's conjecture for fibrations whose general fibres have maximal Clifford index.
In 1985 Xiao Gang proved that the bicanonical system of a complex surface $S$ of general type with $p_2(S)>2$ is not composed of a pencil [Bull. Soc. Math. France, 113 (1985), 23--51]. When in the end of the 80's it was finally proven that $| 2K_S|$ is base point free, whenever $p_g\geq 1$, the part of this theorem concerning surfaces with $p_g\geq 1$ became trivial. In this note a new proof of this theorem for surfaces with $p_g=0$ is presented.
We focus on the positive dimensional fibres of the Prym map $\mathcal{P}_{g,r}$. We present a direct procedure to investigate infinitely many examples of positive dimensional fibres. Such procedure uses families of Galois coverings of the line admitting a 2-sheeted Galois intermediate quotient. Then we generalize to families of Galois coverings of the line admitting a Galois intermediate quotient of higher degree and we show that the higher degree analogue of the aforementioned procedure gives all the known counterexamples to a conjecture by Xiao on the relative irregularity of a fibration.
Let $f: S\longrightarrow B$ be a non-trivial fibration from a complex projective smooth surface $S$ to a smooth curve $B$ of genus $b$. Let $c_f$ the Clifford index of the generic fibre $F$ of $f$. In [arXiv:1401.7502v4] it is proved that the relative irregularity of $f$, $q_f=h^{1,0}(S)-b$ is less than or equal to $g(F)-c_f$. In particular this proves the (modified) Xiao's conjecture: $q_f\le 1+g(F)/2$ for fibrations of general Clifford index. In this short note we assume that the generic fiber of $f$ is a plane curve of degree $d\ge 5$ and we prove that $q_f\le g(F)-c_f-1$. In particular we …
Motivated by a conjecture of Xiao, we study families of coverings of elliptic curves and their corresponding Prym map $Φ$. More precisely, we describe the codifferential of the period map $P$ associated to $Φ$ in terms of the residue of meromorphic $1$-forms and then we use it to give a characterization for the coverings for which the dimension of $\ker(dP)$ is the least possibile. This is useful in order to exclude the existence of non isotrivial fibrations with maximal relative irregularity and thus also in order to give counterexamples to the Xiao's conjecture mentioned above. The first cou…
Let $f:\, S \to B$ be a locally non-trivial relatively minimal fibration of hyperelliptic curves of genus $g\geq 2$ with relative irregularity $q_f$. We show a sharp lower bound on the slope $λ_f$ of $f$. As a consequence, we prove a conjecture of Barja and Stoppino on the lower bound of $λ_f$ as an increasing function of $q_f$ in this case, and we also prove a conjecture of Xiao on the ampleness of the direct image of the relative canonical sheaf if $λ_f<4$.
We review a combinatoric approach to the Hodge Conjecture for Fermat Varieties and announce new cases where the conjecture is true.
This note contains a new proof of a theorem of Gang Xiao saying that the bicanonical map of a surface S of general type is generically finite if and only if the second plurigenus of S is strictly larger than 2. Such properties are also studied for adjoint linear systems |K_S+L|, where L is any divisor with at least 2 linearly independent sections.
We prove that the abundance conjecture for non-uniruled klt pairs in dimension $n$ implies the abundance conjecture for uniruled klt pairs in dimension $n$, assuming the Minimal Model Program in lower dimensions.
In this note we discuss some examples of non torsion and non algebraic cohomology classes for varieties over finite fields. The approach follows the construction of Atiyah-Hirzebruch and Totaro.
We give a reduction of the irregular case for the effective non-vanishing conjecture by virtue of the Fourier-Mukai transform. As a consequence, we reprove that the effective non-vanishing conjecture holds on algebraic surfaces.
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.
Motivated by a conjecture of Xiao, we study supporting divisors of fibred surfaces. On the one hand, after developing a formalism to treat one-dimensional families of varieties of any dimension, we give a structure theorem for fibred surfaces supported on relatively rigid divisors. On the other hand, we study how to produce supporting divisors by constructing a global adjoint map for a fibration over a curve (generalizing the infinitesimal constructions of Collino, Pirola, Rizzi and Zucconi).
A note on Bondal's conjecture
论文证明了 ample Poisson 线丛关联的连接向量场除非 Poisson 结构为零否则不是局部哈密顿的，并以此支持Bondal猜想关于holomorphic Fano Poisson manifold退化 locus维度的猜想。
In this note, we show that, if the Druzkowski mappings $F(X)=X+(AX)^{*3}$, i.e. $F(X)=(x_1+(a_{11}x_1+...+a_{1n}x_n)^3,...,x_n+(a_{n1}x_1+...+a_{nn}x_n)^3)$, satisfies $TrJ((AX)^{*3})=0$, then $rank(A)\leq 1/2(n+δ)$ where $δ$ is the number of diagonal elements of A which are equal to zero. Furthermore, we show the Jacobian Conjecture is true for the Druzkowski mappings in dimension $\leq 9$ in the case $\prod_{i=1}^{n}a_{ii}\neq0$.
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 $$κ(X)\geκ(Y) + κ(X_η).$$
For a projective K3 surface X we introduce the dense triangulated subcategory S^* of the bounded derived category D^b(Coh(X)) of coherent sheaves on X that is generated by spherical objects. For a K3 surface X over \bar Q it is shown that S^* admits a bounded t-structure if and only if the Bloch-Beilinson conjecture holds for X, i.e. CH^2(X)=Z.
A note on an effective bound for the gonality conjecture
本文改进了gonality猜想中的有效界，将线束次数从4g-3降至4g-4，并指出两个例外情形。
In this short note, we will give the key point of the section conjecture of Grothendieck, that is reformulated by monodromy actions. Here, we will also give the result of the section conjecture for algebraic schemes over a number field.
We reduce the Nowicki conjecture on the Weitzenböck derivation of polynomial algebras to well-known problem of the classical invariant theory.
S=T for Shimura Varieties and Moduli Spaces of p-adic Shtukas
本研究借助Scholze的钻石、v-栈理论及Fargues–Scholze几何Satake等价，证明了Xiao–Zhu提出的S=T猜想，并推导出Hodge型志村簇的Eichler–Shimura关系。
In this mostly expository note, we explain a proof of Tate's two conjectures [Tat65] for algebraic cycles of arbitrary codimension on certain products of elliptic curves and abelian surfaces over number fields.
We give a shorter proof to a recent result by Neuberger, in the real case. Our result is essentially an application of the global asymptotic stability Jacobian Conjecture. We also extend some of the results presented in Neuberger.
We prove the weight-monodromy conjecture for varieties which are p-adically uniformized by a product of the Drinfeld upper half spaces. It is an easy consequence of Dat's work on the cohomology complex of the Drinfeld upper half space.
In the article [PR1] {\it On Hrushovski's proof of the Manin-Mumford conjecture} (Proceedings of the ICM 2002), R. Pink and the author gave a short proof of the Manin-Mumford conjecture, which was inspired by an earlier model-theoretic proof by Hrushovski. The proof given in [PR1] uses a difficult unpublished ramification-theoretic result of Serre. It is the purpose of this note to show how the proof given in [PR1] can be modified so as to circumvent the reference to Serre's result. J. Oesterlé and R. Pink contributed several simplifications and shortcuts to this note.
It is shown that every polynomial function $P : \mathbb{C}^2\longrightarrow \mathbb{C}$ with irreducible fibres of same a genus is a coordinate. In consequence, there does not exist counterexamples F = (P,Q) to the Jacobian conjecture such that all fibres of P are irreducible curves of same a genus.
We estimate the proportion of function fields satisfying certain conditions which imply a function-field analogue of the Fontaine-Mazur conjecture. As a byproduct, we compute the fraction of abelian varieties (or even Jacobians) over a finite field which have a rational point of order l.
A note on the transcendental basepoint-free conjecture for Calabi-Yau manifolds
本文证明Calabi-Yau流形的超越无基点猜想在其Beauville-Bogomolov分解的超Kähler因子上成立时即成立，并利用Bakker-Lehn收缩定理在超Kähler流形上给出该猜想成立的一个充分条件。
We show that the $\ell$-adic Tate conjecture for divisors on smooth proper varieties over finitely generated fields of positive characteristic follows from the $\ell$-adic Tate conjecture for divisors on smooth projective surfaces over finite fields. Similar results for cycles of higher co-dimension are given.
The Lang map, namely the universal dominant rational map to a variety of general type, is constructed and briefly discussed in relation with arithmetic conjectures of Harris, Lang and Manin. Existence of the Lang map follows from the additivity of Kodaira dimension, but the fine structure depends on conjectures on birational classification of algebraic varieties. Serious applications of the Lang map are still being searched.
Grothendieck conjectured in the sixties that the even Kunneth projector (with respect to a Weil cohomology theory) is algebraic and that the homological equivalence relation on algebraic cycles coincides with the numerical equivalence relation. In this note we extend these celebrated conjectures from smooth projective schemes to the broad setting of smooth proper dg categories. As an application, we prove that Grothendieck's original conjectures are invariant under homological projective duality. This leads to a proof of Grothendieck's conjectures in the case of intersections of quadrics, line…
Let $F:\Bbb C^n\to\Bbb C^n$ be a polynomial mapping with a non vanishing Jacobian. If the set $S_F$ of non-properness of $F$ is smooth, then $F$ is a surjective mapping. Moreover, the set $S_F$ can not be connected (this is the Nollet-Xavier Conjecture). Additionally, if $n=2$, then the set $S_F$ of non-properness of $F$ cannot be a curve without self-intersections.
A note on the real Jacobian conjecture in degree 7
该研究针对7次实雅可比猜想展开，证明了最高次齐次部分为特定形式的7次多项式对应的实雅可比猜想成立，同时排除了一类非典型雅可比对的存在性。
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。
The aim of this paper is to get a complete list of positive definite symmetric matrices with integer entries $\a&b\b&d\$ such that all complex solutions to the system of equations $1-x_1=x_1^ax_2^b\ 1-x_2=x_1^bx_2^d$ are real. This result is related to Nahm's conjecture in rank 2 case.
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