Matthias Schuett
We prove that the maximal singular fibres of elliptic K3 surfaces have type I_19 and I_14* unless the characteristic of the ground field is 2. In characteristic 2, the maximal singular fibres are I_18 and I_13*. The paper supplements work of Shioda, published in 2003 and 2005.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We generalise a method of Xiao Gang to construct 'prototypes' of fibred surfaces with maximal irregularity without being a product. This enables us, in the case of fibre genus g=3 to describe the possible singular fibres and to calculate the invariants of these surfaces. We also prove structure theorems on the moduli space for fibred surfaces with fibre genus g=2 and g=3.
In this work we describe a construction of semistable fibrations over an elliptic curve with one unique singular fibre and we give effective examples using monodromy of curves.
We study deformation theory of elliptic fibre bundles over curves in positive characteristics. As applications, we give examples of non-liftable elliptic surfaces in charactertic two and three, which answers a question of Katsura and Ueno. Also, we construct a class of elliptic fibrations, whose liftability is equivalent to a conjecture of F. Oort concerning the liftability of automorphisms of curves. Finally, we classify deformations of bielliptic surfaces.
We determine and list all possible configurations of singular fibres on rational elliptic surfaces in characteristic three. In total, we find that 267 distinct configurations exist. This result complements Miranda and Persson's classification in characteristic zero, and Lang's classification in characteristic two.
We study threefolds fibred by Kummer surfaces associated to products of elliptic curves, that arise as resolved quotients of threefolds fibred by certain lattice polarized K3 surfaces under a fibrewise Nikulin involution. We present a general construction for such surfaces, before specializing our results to study Calabi-Yau threefolds arising as resolved quotients of threefolds fibred by mirror quartic K3 surfaces. Finally, we give some geometric properties of the Calabi-Yau threefolds that we have constructed, including expressions for Hodge numbers.
Rational elliptic surfaces with six singular double fibres
本文研究具有六个奇异双纤维的有理椭圆面，通过多种视角进行分类并给出构造方法。
We consider an elliptic surface $π: \mathcal{E}\rightarrow \mathbb{P}^1$ defined over a number field $k$ and study the problem of comparing the rank of the special fibres over $k$ with that of the generic fibre over $k(\mathbb{P}^1)$. We prove, for a large class of rational elliptic surfaces, the existence of infinitely many fibres with rank at least equal to the generic rank plus two.
We classify singular fibres over general points of the discriminant locus of projective complex Lagrangian fibrations on 4-dimensional holomorphic symplectic manifolds. The singular fibre F is the following either one: F is isomorphic to the product of an elliptic curve and a Kodaira singular fibre up to finite unramified covering or F is a normal crossing variety consisting of several copies of a minimal elliptic ruled surface of which the dual graph is Dynkin diagram of type A_n, $\tilde{A_n}$, $\tilde{D_n}$ or D_n.
We explicitly determine the elliptic K3 surfaces with a maximal singular fibre. If the characteristic of the ground field is different from 2, for each of the two possible maximal fibre types, $I_{19}$ and $I^*_{14}$, the surface is unique. In characteristic 2 the maximal fibre types are $I_{18}$ and $I^*_{13}$, and there exist two (resp. one) one-parameter families of such surfaces.
Using Lagrangian Floer theory, we study the tropical geometry of K3 surfaces with general singular fibres. In particular, we give the local models for the type $I_n$, $II$, $III$ and $IV$ singular fibres in the Kodaira's classification and generalize the correspondence theorem between open Gromov-Witten invariants/tropical discs counting to these cases.
We show that simultaneous log resolutions of simply elliptic singularities can be constructed inside suitable stacks of principal bundles over elliptic curves. In particular, we give a direct geometrical construction of del Pezzo surfaces from the corresponding exceptional simple algebraic groups.
Periods of fibre products of elliptic surfaces and the Gamma conjecture
本文提出计算$\mathbb P^1$上椭圆曲面纤维积同调基、相交积和周期矩阵的算法，用于研究由此得到的Calabi-Yau三维流形的Gamma猜想，发现适用于105个纤维积算子及Calabi-Yau数据库四阶算子的公式，并提供SageMath实现。
Locally non-trivial fibred surfaces with maximal unitary rank
该文研究具有最大酉秩$u_f=g-1$的局部非平凡纤维化曲面，证明强斜率不等式，结合三角曲线结果给出$g=6$、$u_f=5$情形的新约束，包括曲面指数严格为正及相对典范除子类的强限制。
We define a monodromy homomorphism for irreducible families of regular elliptic fibrations which takes values in the mapping class group of a punctured sphere. We compute the monodromy for elliptic fibrations only which contain no singular fibres of types other than I_1 and I*_0. There are maximal groups, which can be the monodromy groups of algebraic, resp. differentiable families of elliptic surfaces, and we give an algebraic criterion for the equality of both groups which we can check to apply in case the number of I_1 fibres is at most 6.
We study non-isotrivial projective families of elliptic surfaces of Kodaira dimension one, over complex projective curves. If the base is an elliptic curve, we show that the family must have a singular fibre, and that over the projective line it must have at least three singular fibres. Similar results, for families of surfaces of general type, have been obtained by Migliorini and Kovács, and they are well-known for projective families of surfaces of Kodaira dimension zero. Revised version: We corrected some minor errors and ambiguities, and we completed the list of references.
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.
Geometric variations of local systems are families of variations of Hodge structure; they typically correspond to fibrations of Kähler manifolds for which each fibre itself is fibred by codimension one Kähler manifolds. In this article, we introduce the formalism of geometric variations of local systems and then specialize the theory to study families of elliptic surfaces. We interpret a construction of twisted elliptic surface families used by Besser-Livné in terms of the middle convolution functor, and use explicit methods to calculate the variations of Hodge structure underlying the univers…
Jacobian elliptic fibrations on K3s with a non-symplectic automorphism of order 3
本文依据3阶非辛自同构对纤维的作用，结合Kneser–Nishiyama方法分类相应K3曲面上的Jacobian椭圆纤维化，并给出可约纤维类型与σ不变Weierstrass方程。
We prove a relative version of the theorem of Cho, Miyaoka and Shepherd-Barron: a Mori fibre space of maximal length is birational to a projective bundle.
In this paper we investigate two stratifications of the moduli space of elliptically fibred K3 surfaces. The first comes from Shimada's classification of connected components of elliptically fibred K3 surfaces and is closely related to the root lattice of the fibration. The second is the monodromy stratification defined by Bogomolov, Petrov and Tschinkel. The main result of the paper is a classification of all positive-dimensional ambi-typical strata, that is strata which are both Shimada root strata and monodromy strata. We further discuss the connection with moduli spaces of lattice-polarise…
A new class of examples of surfaces with maximal Picard number is constructed. These carry pencils of genus two or three curves such their Jacobian fibrations are isogenous to fibre products of elliptic modular surfaces.
Fibred surfaces and their unitary rank
本文研究亏格至少为2的复纤维化曲面的酉秩，证明多个新斜率不等式，并由此得到非等平凡纤维化酉秩的Xiao型上界及Coleman-Oort猜想方向的新约束。
The platonic elliptic surfaces
本文构造了附属于柏拉图立体的特殊有理椭圆曲面，研究其代数几何性质等，发现需改进Dwork展开以确定其纤维化纤维的欧拉因子。
We calculate the Tate-Shafarevich group of an elliptic three-fold $f:X\rightarrow S$ when $X$ and $S$ are regular and $f$ is flat, relating it to the Brauer group of $X$ and $S$. We show that given certain hypotheses on $f$, the Tate-Shafarevich group has the interpretation of isomorphism classes of elliptic curves over the function field of $S$ which have the same jacobian as the generic fibre of $f$, and for which there exists a relatively minimal model which has no multiple fibres. We use this to give examples of elliptic fibrations with isolated multiple fibres, and also to give a new coun…
Elliptic fibrations on toric $K3$ hypersurfaces and mirror symmetry derived from Fano polytopes
本文确定了由Fano多面体导出的环面$K3$超曲面的Néron-Severi格，通过引入特定椭圆纤维化并证明其生成元，从而证明了Dolgachev猜想。
Degenerations and Fibrations of K3 Surfaces: Lattice Polarisations and Mirror Symmetry
本文研究K3曲面Tyurin退化及椭圆纤维化的格极化理论，建立兼容现有定义的格极化概念，并探讨两种设置间的镜像对称对应。
A fibration is said to be isotrivial if all of its smooth fibres are isomorphic to a single fixed variety. We classify the elliptic K3 surfaces that are isotrivial, and use them to construct Lagrangian fibrations that are isotrivial. We then modify the construction to produce new examples of holomorphic symplectic orbifolds, that also admit isotrivial Lagrangian fibrations.
K. Konno proved a slope equality for fibred surfaces with fibres of odd genus and general fibre of maximal gonality. More precisely he found a relation between the invariants of the fibration and certain weights of special fibres (called the Horikawa numbers). We give an alternative and more geometric proof in the case of a genus 5 fibration, under generality assumptions. In our setting we are able to prove that the fibre with positive Horikawa numbers are precisely the trigonal ones, we compute their weights explicitly and thus we exhibit explicit examples of regular surfaces with assigned in…
We consider the maximal number of arbitrary points in a special fibre that can be simultaneously approached by points in one sequence of general fibres. Several results about this topological invariant and their applications describe the structure of holomorphic maps. In particular, we get a lower bound on the number of points in the general fibre of a generically finite map.
In this paper we want to study the non-Kodaira fibres in a smooth equidimensional elliptic threefold. If the morphism to the Weierstrass model of the fibration is crepant, then we can locate the non-Kodaira fibres and give a description of their structure. In particular, they lie over the singular points of the (reduced) discriminant of the fibration and are contraction of a Kodaira fibre, whose type can be predicted using Tate's algorithm.
Shioda described in his article from 1986 a method to compute the Lefschetz number of a Delsarte surface. In one of his examples he uses this method to compute the rank of an elliptic curve over k(t). In this article we find all elliptic curves over k(t) for which his method is applicable. For each of these curves we also compute the Mordell-Weil rank.
Using elliptic structures, we show that any supersingular K3 surface of Artin invariant $1$ in characteristic $p \not= 5$, $7$, $13$ has an automorphism the entropy of which is the natural logarithm of a Salem number of degree $22$.
We study the maximal Salem degree of automorphisms of K3 surfaces via elliptic fibrations. By generalizing \cite{EOY14}, we establish a characterization of such maximum in terms of elliptic fibrations with infinite automorphism groups. As an application, we show that any supersingular K3 surface in odd characteristic has an automorphism the entropy of which is the natural logarithm of a Salem number of degree $22$.
Let $X$ be a surface of general type with maximal Albanese dimension: if $K_X^2<\frac{9}{2}χ(\mathcal{O}_X)$, one has $K_X^2\geq 4χ(\mathcal{O}_X)+4(q-2)$. We give a complete classification of surfaces for which equality holds for $q(X)\geq 3$: these are surfaces whose canonical model is a double cover of a product elliptic surface branched over an ample divisor with at most negligible singularities which intersects the elliptic fibre twice. We also prove, in the same hypothesis, that a surface $X$ with $K_X^2\neq 4χ(\mathcal{O}_X)+4(q-2)$ satisfies $K_X^2\geq 4χ(\mathcal{O}_X)+8(q-2)$ and we …
We study the elliptic modular surface attached to the commutator subgroup of the modular group. This has an elliptic curve as base and only one singular fibre. We employ an algebraic approach and then consider some arithmetic questions.
We explicitly construct Brill--Noether general $K3$ surfaces of genus $4,6$ and $8$ having the maximal number of elliptic pencils of degrees $3, 4$ and $5$, respectively, and study their moduli spaces and moduli maps to the moduli space of curves. As an application we prove the existence of Brill--Noether general $K3$ surfaces of genus $4$ and $6$ without stable Lazarsfeld--Mukai bundles of minimal $c_2$.
Explicit birational models of marked elliptic surfaces of relative degree two
本文针对带双截面或二次相对极化、含二重纤维的椭圆曲面，提出两种双有理模型的显式构造方法，并将其简洁描述应用于指数2的Halphen曲面与Enriques曲面。
A class of geometrically elliptic fibrations by plane projective quartic curves
本文研究特征2下算术亏格3、几何亏格1的非超椭圆曲线纤维化，在特定奇点假设下给出双有理等价的完整分类，证明其可被椭圆纤维化覆盖且覆盖映射具特定双有理与纯不可分性质。
Let $E$ be an elliptic surface over the curve $C$, defined over a number field $k$, let $P$ be a section of $E$, and let $\ell$ be a rational prime. For any non-singular fibre $E_t$, we bound the number of points $Q$ on $E_t$ of (algebraic) degree at most $D$ over $k$, such that $\ell^n Q=P_t$, for some $n\geq 1$. The bound obtained depends only on $\ell$, the surface and section in question, $D$, and the degree $[k(t):k]$; that is, it is uniform across all fibres of bounded degree. In special cases, we obtain more specific, in some instances sharp, bounds.
We present a complete list of extremal elliptic K3 surfaces. There are altogether 325 of them. The first 112 coincides with Miranda-Persson's list for semi-stable ones. The data include the transcendental lattice which determines uniquely the K3 surface by a result of Shioda and Inose, the singular fibre type and the Mordell Weil group. As an application, we give a sufficient condition for the topological fundamental group of complement to an ADE-configuration of smooth rational curves on a K3 surface to be trivial.
Let $X$ be a surface of general type with maximal Albanese dimension over an algebraically closed field of characteristic greater than two: we prove that if $K_X^2<\frac{9}{2}χ(\mathcal{O}_X)$, one has $K_X^2\geq 4χ(\mathcal{O}_X)+4(q-2)$. Moreover we give a complete classification of surfaces for which equality holds for $q(X)\geq 3$: these are surfaces whose canonical model is a double cover of a product elliptic surface branched over an ample divisor with at most negligible singularities which intersects the elliptic fibre twice. In addition we expose a similar partial result over algebraic…
Given a non-singular variety with a K3 fibration f : X --> S we construct dual fibrations Y --> S by replacing each fibre X_s of f by a two-dimensional moduli space of stable sheaves on X_s. In certain cases we prove that the resulting scheme Y is a non-singular variety and construct an equivalence of derived categories of coherent sheaves Φ: D(Y) --> D(X). Our methods also apply to elliptic and abelian surface fibrations. As an application we show how the equivalences Φidentify certain moduli spaces of stable bundles on elliptic threefolds with Hilbert schemes of curves.
Maximal singular curves over finite fields from elliptic and hyperelliptic curves
本文构造有限域上达到Aubry-Perret界的极大奇异曲线族，利用椭圆/超椭圆曲线的线性投影和Gorenstein曲线理论，并确定所得曲线的gonality。
Given an elliptic surface over a number field, we study the collection of fibres whose Mordell-Weil rank is greater than the generic rank. Under suitable assumptions, we show that this collection is not thin. Our results apply to quadratic twist families and del Pezzo surfaces of degree $1$.
We prove that every non-trivial structure of a rationally connected fibre space (and so every structure of a Mori-Fano fibre space) on a general (in the sense of Zariski topology) hypersurface of degree $M$ in the $(M+1)$-dimensional projective space for $M\geq 14$ is given by a pencil of hyperplane sections. In particular, the variety $V$ is non-rational and its group of birational self-maps coincide with the group of biregular automorphisms and is therefore trivial. The proof is based on the techniques of the method of maximal singularities and the inversion of adjunction.
Normal Forms and Tyurin Degenerations of K3 Surfaces Polarised by a Rank 18 Lattice
本文研究由秩18格极化的K3曲面的射影II型退化，构造了一系列通过单曲线flop相互关联的射影例子，并证明该列表完备，说明射影性对中心纤维分支曲面类型不构成实质限制。
Cohomology of ideals in elliptic surface singularities
本文研究了二维正常奇点的正常还原数，证明了椭圆奇点的正常还原数为二，并探讨了非有理奇点的Gorenstein性质与$p_g$-理想的关系。
We prove that Godeaux--Reid surfaces with torsion group Z/3 have topological fundamental group Z/3. For this purpose, we describe degenerations to stable KSBA surfaces with one 1/4(1,1) singularity, whose minimal resolution are elliptic fibrations with two multiplicity 3 fibres and one I_4 singular fibre. We study special such degenerations which have an involution, describing the corresponding Campedelli double plane construction. We also find some stable rational degenerations, some of which have more singularities, and one of which has a single 1/9(1,2) singularity, the minimal possible ind…
Torsion of elliptic curves with rational $j$-invariant over the maximal elementary abelian 2-extension of $\mathbb{Q}$
本文针对定义在有理数域所有二次扩张合成域上、j不变量为非0或1728的有理数的椭圆曲线，完成了对其可能挠子群结构的分类。
We show that the Davenport-Stothers inequality from characteristic 0 fails in any characteristic p>3. The proof uses elliptic surfaces over the projective line and inseparable base change. We then present adjusted inequalities. These follow from results of Pesenti-Szpiro. For characteristic 2 and 3, we achieve a similar result in terms of the maximal singular fibres of elliptic surfaces over the projective line. Our ideas are also related to supersingular surfaces (in Shioda's sense).
On the Chow group of elliptic surfaces over number fields
本文研究数字段上椭圆面的Chow群，证明其核是一个有限生成群。
Brauer groups of conic bundles over elliptic curves
本文研究数域上椭圆曲线的正则圆锥丛的Brauer群，在奇异纤维位于2-可除点且分歧域同构的条件下给出显式计算，并应用于一类Châtelet型曲面，进而分析实位点产生的Brauer-Manin障碍对弱逼近的影响。
We consider $M$-theory compactified on $T^4 \times T^2$ and describe the count of spinning $1/8$-BPS states. This refines the classic count of Maldacena-Moore-Strominger in the physics literature and the recent mathematical work of Bryan-Oberdieck-Pandharipande-Yin, which studied reduced Donaldson-Thomas invariants of abelian surfaces and threefolds. As in previous work on $K3 \times T^2$ compactification, we track angular momenta under both the $SU(2)_L$ and $SU(2)_R$ factors in the 5d little group, providing predictions for the relevant motivic curve counts.
We describe a method for doing computations with Orlov's equivalence between the bounded derived category of certain hypersurfaces and the stable category of graded matrix factorisations of the polynomials describing these hypersurfaces. In the case of a smooth elliptic curve over an algebraically closed field we describe the indecomposable graded matrix factorisations of rank one. Since every indecomposable Maximal Cohen-Macaulay module over the completion of a smooth cubic curve is gradable, we obtain explicit descriptions of all indecomposable rank one matrix factorisations of such potentia…
Given a minimal surface equipped with a generically finite map to an Abelian variety, we give an optimal bound on the canonical degree of a rational or an elliptic curve. As a corollary, we obtain the finiteness of rational and elliptic curves on any surface of general type with two linearly independent regular one forms.
In this paper, we explicitly construct the smooth compact base threefold for the elliptic Calabi-Yau fourfold with the largest known $h^{1,1}=303\,148$. It is generated by blowing up a smooth toric "seed" base threefold with $(E_8,E_8,E_8)$ collisions. The 4d F-theory compactification model over it has the largest geometric gauge group, $E_8^{2\,561}\times F_4^{7\,576}\times G_2^{20\,168}\times SU(2)^{30\,200}$, and the largest number of axions, $181\,820$, in the known 4d $\mathcal{N}=1$ supergravity landscape. We also prove that there are at least $1100^{15\,048}\approx 7.5\times 10^{45\,766…
We consider the 33 conjugacy classes of genus zero, torsion-free modular subgroups, computing ramification data and Grothendieck's dessins d'enfants. In the particular case of the index 36 subgroups, the corresponding Calabi-Yau threefolds are identified, in analogy with the index 24 cases being associated with K3 surfaces. In a parallel vein, we study the 112 semi-stable elliptic fibrations over P^1 as extremal K3 surfaces with six singular fibres. In each case, a representative of the corresponding class of subgroups is identified by specifying a generating set for that representative.
Given two semistable, non potentially isotrivial elliptic surfaces over a curve $C$ defined over a field of characteristic zero or finitely generated over its prime field, we show that any compatible family of effective isometries of the N{é}ron-Severi lattices of the base changed elliptic surfaces for all finite separable maps $B\to C$ arises from an isomorphism of the elliptic surfaces. Without the effectivity hypothesis, we show that the two elliptic surfaces are isomorphic. We also determine the group of universal automorphisms of a semistable elliptic surface. In particular, this includes…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。