In this note we show how to find the stable model of a one-parameter family of elliptic surfaces with sections.
More specifically, we perform the log Minimal Model Program in an explicit manner by means of toric geometry, in each such one parameter family. This way we obtain an explicit combinatorial description of the surfaces that may occur at the boundary of moduli (as well as a new proof of the completeness of the moduli space)
In this article, we study the geometry of plane curves obtained by three sections and another section given as their sum on certain rational elliptic surfaces. We make use of Mumford representations of semi-reduced divisors in order to study the geometry of sections. As a result, we are able to give new proofs for some classical results on singular plane quartics and their bitangent lines.
We classify elliptic K3 surfaces in characteristic $p$ with $p^n$-torsion sections. For $p^n\geq3$ we verify conjectures of Artin and Shioda, compute the heights of their formal Brauer groups, as well as Artin invariants and Mordell--Weil groups in the supersingular cases.
In this article, examples of Zariski pairs $(B_1, B_2)$ satisfying the following condition are given: (i) $°B_1 = °B_2 = 7$. (ii) Irreducible components of $B_i$ $(i = 1, 2)$ are lines and conics. (iii) Singularities of $B_i$ $(i = 1, 2)$ are nodes, tacnodes and ordinary triple points. In order to construct $B_i$ ($i = 1, 2$), we make use of geometry of sections of rational elliptic surfaces and their group structure. Dihedral covers play important roles to distinguish the topology of $({\mathbb P}^2, B_i)$ $(i = 1, 2)$.
We investigate Calabi--Yau three folds which are small resolutions of fiber products of elliptic surfaces with section admitting reduced fibers. We start by the classification of all fibers that can appear on such varieties. Then, we find formulas to compute the Hodge numbers of obtained three folds in terms of the types of singular fibers of the elliptic surfaces. Next we study Kummer fibrations associated to these fiber products.
We give a geometrical criterion to determine when a quaternion algebra over the function field of a stable elliptic surface X is an Azumaya algebra over X.
Let $f \colon X \to B$ be a nonisotrivial complex elliptic surface and let $\mathcal{D} \subset X$ be an integral divisor dominating $B$. We study finiteness related properties of generalized $(S, \mathcal{D})$-integral sections $σ\colon B \to X$ of $X$. These integral sections $σ$ correspond to rational points in $A(K)$ which satisfy the set-theoretic condition $f ( σ(B) \cap \mathcal{D})\subset S$, where $S \subset B$ is an arbitrary given subset. For $S \subset B$ finite, the set of $(S, \mathcal{D})$-integral sections of $X$ is finite by the well-known Siegel theorem. In this article, we e…
We prove that there are finitely many families, up to isomorphism in codimension one, of elliptic Calabi-Yau manifolds $Y\rightarrow X$ with a rational section, provided that $\dim(Y)\leq 5$ and $Y$ is not of product-type. As a consequence, we obtain that there are finitely many possibilities for the Hodge diamond of such manifolds. The result follows from log birational boundedness of klt pairs $(X, Δ)$ with $K_X+Δ$ numerically trivial and not of product-type, in dimension at most $4$.
We show that fiberwise stable vector bundles are preserved by relative Fourier-Mukai transforms between elliptic threefolds with relative Picard number one. Using these bundles we define new invariants of elliptic fibrations, and we relate the invariants of a space with those of a relative moduli space of stable sheaves on it. As a byproduct, we calculate the intersection form of a certain new example of an elliptic Calabi-Yau threefold.
Given a torsion section of a semistable elliptic surface, we prove equidistribution results for the components of singular fibers which are hit by the section, and for the root of unity (identifying the zero component with ${\Bbb C}$) which is hit by the section in case the section hits the zero component.
We give explicit generators for subgroups of finite index of the Mordell-Weil groups of several families of elliptic surfaces introduced by Masato Kuwata.
We consider a rational elliptic surface with a relatively minimal fibration. We compute the number of integral sections in the above rational elliptic surface. As an application, we obtain an estimate of polynomial solutions of some equations.
It is known that the main component of the Kontsevich's moduli space of elliptic stable maps is singular. There are two different desingularizations. One is Vakil-Zinger's desingularization and the other is the moduli space of logarithmic stable maps. When the degree is less then or equal to 3 and the target is $\mathbb{P}^{n}$, we show that the moduli space of logarithmic stable maps can be obtained by blowing up Vakil-Zinger's desingularization.
Let S be a torsion section of an elliptic surface with only I_n fibers. This article addresses the question: which components of singular fibers can S pass through? We give necessary criteria for the "component numbers", and show an equidistribution result for torsion sections of prime order.
Desingularized fiber products of semi-stable, non-isotrivial jacobian elliptic surfaces with vanishing third Betti number are classified. Such varieties may play a role in the study of supersingular threefolds, of the deformation theory of varieties with trivial canonical bundle, and of the arithmetic degenerations of rigid Calabi-Yau threefolds.
Let $M$ be the moduli space of rank $2$ stable bundles with fixed determinant of degree $1$ on a smooth projective curve $C$ of genus $g\ge 2$. When $C$ is generic, we show that any elliptic curve on $M$ has degree (respect to anti-canonical divisor $-K_M$) at least 6, and we give a complete classification for elliptic curves of degree $6$. Moreover, if $g>4$, we show that any elliptic curve passing through the generic point of $M$ has degree at least $12$. We also formulate a conjecture for higher rank.
Given an elliptic surface $\mathcal{E}\to\mathcal{C}$ over a field $k$ of characteristic zero equipped with zero section $O$ and another section $P$ of infinite order, we give a simple and explicit upper bound on the number of points where $O$ is tangent to a multiple of $P$.
Let $f \colon X \to B$ be a complex elliptic surface and let $\DD \subset X$ be an integral divisor dominating $B$. It is well-known that the Parshin-Arakelov theorem implies the Mordell conjecture over complex function fields by a beautiful covering trick of Parshin. In this article, we construct a similar map in the context of $(S, \DD)$-integral points on elliptic curves over function fields to obtain a new proof of certain uniform finiteness results on the number of $(S, \DD)$-integral points. A second new proof is also given by establishing a uniform bound on the canonical height by means…
Let $M$ be the moduli space of rank 3 stable bundles with fixed determinant of degree 1 on a smooth projective curve of genus $g\geq 2$. When $C$ is generic, we show that any essential elliptic curve on $M$ has degree (respect to anti-canonical divisor $-K_{M}$) at least 6, and we give a complete classification for elliptic curves of degree 6, which is not in conformity with Sun's Conjecture. Moreover, if $g>12$, we show that any elliptic curve passing through the generic point of $M$ has degree at least 18.
We show that a general elliptic K3 surface with a section is determined uniquely by its discriminant, which is a configuration of 24 points on the projective line. It follows that a general elliptic K3 surface with a section can be reconstructed from its Gromov-Hausdorff limit as the volume of the fiber goes to zero.
We describe two geometrically meaningful compactifications of the moduli space of elliptic K3 surfaces via stable slc pairs, for two different choices of a polarizing divisor, and show that their normalizations are two different toroidal compactifications of the moduli space, one for the ramification divisor and another for the rational curve divisor. In the course of the proof, we further develop the theory of integral affine spheres with 24 singularities. We also construct moduli of rational (generalized) elliptic stable slc surfaces of types ${\bf A_n}$ ($n\ge1$), ${\bf C_n}$ ($n\ge0$) and …
We consider a relative Fourier-Mukai transform defined on elliptic fibrations over an arbitrary normal base scheme. This is used to construct relative Atiyah sheaves and generalize Atiyah's and Tu's results about semistable sheaves over elliptic curves to the case of elliptic fibrations. Moreover we show that this transform preserves relative (semi)stability of sheaves of positive relative degree.
On a Weierstraß elliptic surface $X$, we define a `limit' of Bridgeland stability conditions, denoted $Z^l$-stability, by varying the polarisation in the definition of Bridgeland stability along a curve in the ample cone of $X$. We show that a slope stable torsion-free sheaf of positive (twisted) degree or a slope stable locally free sheaf is taken by a Fourier-Mukai transform on $D^b(X)$ to a $Z^l$-stable object, while a $Z^l$-semistable object of nonzero fiber degree can be modified so that its inverse Fourier-Mukai transform is a slope semistable torsion-free sheaf.
We analyze K3 surfaces admitting an elliptic fibration $E$ and a finite group $G$ of symplectic automorphisms preserving this elliptic fibration. We construct the quotient elliptic fibration $E/G$ comparing its properties to the ones of $E$. We show that if $E$ admits an $n$-torsion section, its quotient by the group of automorphisms induced by this section admits again an $n$-torsion section. Considering automorphisms coming from the base of the fibration, we can describe the Mordell--Weil lattice of a fibration described by Kloosterman. We give the isometries between lattices described by th…
On the product elliptic threefold $X = C \times S$ where $C$ is an elliptic curve and $S$ is a K3 surface of Picard rank 1, we define a notion of limit tilt stability, which satisfies the Harder-Narasimhan property. We show that under the Fourier-Mukai transform $Φ$ on $D^b(X)$ induced by the classical Fourier-Mukai transform on $D^b(C)$, a slope stable torsion-free sheaf satisfying a vanishing condition in codimension 2 (e.g. a reflexive sheaf) is taken to a limit tilt stable object. We also show that a limit tilt semistable object on $X$ is taken by $Φ$ to a slope semistable sheaf, up to mod…
We study semistable pairs on elliptic K3 surfaces with a section: we construct a family of moduli spaces of pairs, related by wall crossing phenomena, which can be studied to describe the birational correspondence between moduli spaces of sheaves of rank 2 and Hilbert schemes on the surface. In the 4-dimensional case, this can be used to get the isomorphism between the moduli space and the Hilbert scheme described by Friedman.
Let E be an elliptic curve over the function field Q(t). Suppose that for every number field L\not=Q and every element tau\in L such that the specialization E_tau is smooth, the curve E_tau has a non-trivial torsion point over L. We show that E has a non-trivial torsion point over Q(t). This provides evidence in support of a question of Graber-Harris-Mazur-Starr on rational pseudo-sections of arithmetic surjective morphisms.
We prove that a very general elliptic surface $\mathcal{E}\to\mathbb{P}^1$ over the complex numbers with a section and with geometric genus $p_g\ge2$ contains no rational curves other than the section and components of singular fibers. Equivalently, if $E/\mathbb{C}(t)$ is a very general elliptic curve of height $d\ge3$ and if $L$ is a finite extension of $\mathbb{C}(t)$ with $L\cong\mathbb{C}(u)$, then the Mordell-Weil group $E(L)=0$.
We consider elliptic surfaces $\mathcal{E}$ over a field $k$ equipped with zero section $O$ and another section $P$ of infinite order. If $k$ has characteristic zero, we show there are only finitely many points where $O$ is tangent to a multiple of $P$. Equivalently, there is a finite list of integers such that if $n$ is not divisible by any of them, then $nP$ is not tangent to $O$. Such tangencies can be interpreted as unlikely intersections. If $k$ has characteristic zero or $p>3$ and $\mathcal{E}$ is very general, then we show there are no tangencies between $O$ and $nP$. We apply these res…
A degree one del Pezzo surface is the blowup of P^2 at 8 general points. By the classical Cayley-Bacharach Theorem, there is a unique 9th point whose blowup produces a rational elliptic surface with a section. Via this relationship, we construct a stable pair compactification of the moduli space of anti-canonically polarized degree one del Pezzo surfaces.
Given a rational elliptic surface X over an algebraically closed field, we investigate whether a given natural number k can be the intersection number of two sections of X. If not, we say that k a gap number. We try to answer when gap numbers exist, how they are distributed and how to identify them. We use Mordell-Weil lattices as our main tool, which connects the investigation to the classical problem of representing integers by positive-definite quadratic forms.
We give a classification of the log canonical models of elliptic surface pairs consisting of an elliptic fibration, a section, and a weighted sum of marked fibers. In particular, we show how the log canonical models depend on the choice of the weights. We describe a wall and chamber decomposition of the space of weights based on how the log canonical model changes. In addition, we give a generalized formula for the canonical bundle of an elliptic surface with section and marked fibers. This is the first step in constructing compactifcations of moduli spaces of elliptic surfaces using the minim…
We characterize the subscheme of the moduli space of torsion-free sheaves on an elliptic surface which are stable of relative degree zeero (with respect to a polarization of type aH+bf, H being the section and f the elliptic fibre) which is isomorphic, via the relative Fourier-Mukai transform, with the relative compactified Simpson Jacobian of the family of those curves D in the surface which are flat over the base of the elliptic fibration. This generalizes and completes earlier constructions due to Friedman, Morgan and Witten. We also study the relative moduli scheme of sheaves whose restric…
This is the third paper in a series of work on weighted stable elliptic surfaces - elliptic fibrations with section and marked fibers weighted between zero and one. Motivated by Hassett's weighted pointed stable curves, we use the log minimal model program to construct compact moduli spaces parameterizing these objects. Moreoever, we show that the domain of weights admits a wall and chamber structure, we describe the induced wall crossing morphisms on the moduli spaces as the weight vector varies, and describe the surfaces that appear on the boundary of the moduli space. The main technical res…
We focus on a class of Weierstrass elliptic threefolds that allows the base of the fibration to be a Fano surface or a numerically $K$-trivial surface. We define the notion of limit tilt stability, and show that the Fourier-Mukai transform of a slope stable torsion-free sheaf satisfying a vanishing condition in codimension 2 (e.g. a reflexive sheaf) is a limit tilt stable object. We also show that the inverse Fourier-Mukai transform of a limit tilt semistable object of nonzero fiber degree is a slope semistable torsion-free sheaf, up to modification in codimension 2.
We study elliptic K3 surfaces with Mordell Weil rank 0, and which has a 2-torsion section $σ$ such that the translation by $σ$ gives a Shioda-Inose structure.
This paper is concerned with the construction of extremal elliptic K3 surfaces. It gives a complete treatment of those fibrations which can be derived from rational elliptic surfaces by easy manipulations of their Weierstrass equations. In particular, this approach enables us to find explicit equations for 38 semi-stable extremal elliptic K3 fibrations, 32 of which are indeed defined over the rationals. They are realized as pull-back of non-semi-stable extremal rational elliptic surfaces via base change. This is related to work of J. Top and N. Yui which exhibited the same procedure for the se…
On a Weierstraß elliptic surface $X$, we define a `limit' of Bridgeland stability conditions, denoted as $Z^l$-stability, by moving the polarisation towards the fiber direction in the ample cone while keeping the volume of the polarisation fixed. We describe conditions under which a slope stable torsion-free sheaf is taken by a Fourier-Mukai transform to a $Z^l$-stable object, and describe a modification upon which a $Z^l$-semistable object is taken by the inverse Fourier-Mukai transform to a slope semistable torsion-free sheaf. We also study wall-crossing for Bridgeland stability, and show th…
Rational elliptic surfaces with six singular double fibres
Università di Roma Tor Vergata（罗马托尔维加塔大学）；Università di Bologna（博洛尼亚大学）；University of Pennsylvania（宾夕法尼亚大学）；Colorado State University（科罗拉多州立大学）；Università Roma Tre（罗马第三大学）；École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
In this paper we construct new examples of stable bundles of rank 2 of small degree with 4 sections on a smooth irreducible curve of maximal Clifford index. The corresponding Brill-Noether loci have negative expected dimension of arbitrarily large absolute value.
This paper deals with a study of the rational elliptic surfaces whose $J$-invariant functions are of degree one. Almost all of these elliptic surfaces have four singular fibers, while the remaining surfaces have only three singular fibers. The moduli space of these elliptic surfaces is canonically isomorphic to the projective line by taking the $J$-values for a certain fixed type of singular fibers. Over the moduli space, we discuss our elliptic surfaces, and investigate how their sections are described by the parameter of the moduli space. By using a covering space of the moduli, we construct…
We discuss the relation between the graded stable derived category of a hypersurface and that of its hyperplane section. The motivation comes from the compatibility between homological mirror symmetry for the Calabi-Yau manifold defined by an invertible polynomial and that for the singularity defined by the same polynomial.
We construct stable bundle extensions on elliptically fibered Calabi-Yau threefolds. We show that these bundles can solve the topological anomaly constraint in heterotic string theory without the need of invoking background fivebranes.
We shall consider sections of an elliptic scheme $\mathcal{E}$ over a(n affine) base curve $B$, and study the points of $B$ where the section takes a torsion value. In particular, we shall relate the distribution in $B$ of these points with the canonical height of the section, proving an integral formula involving a measure on $B$ coming from the so-called Betti map of the section. We shall show that this measure is the same appearing in dynamical issues related to the section. This analysis will also involve the multiplicity with which a torsion value is attained, which is an independent prob…
To count bundles on curves, we study zetas of elliptic curves and their zeros. There are two types, i.e., the pure non-abelian zetas defined using moduli spaces of semi-stable bundles, and the group zetas defined for special linear groups. In lower ranks, we show that these two types of zetas coincide and satisfy the Riemann Hypothesis. For general cases, exposed is an intrinsic relation on automorphism groups of semi-stable bundles over elliptic curves, the so-called counting miracle. All this, together with Harder-Narasimhan, Desale-Ramanan and Zagier's result, gives an effective way to coun…
We prove new sharp asymptotic for counting the semistable elliptic curves with two marked Weierstrass points at $\infty$ and $0$ and also the cases where $0$ is a 2-torsion or a 3-torsion marked Weierstrass point over $\mathbb{F}_q(t)$ by the bounded height of discriminant $Δ(X)$. We consider the motivic probabilities over any basefield $K$ with $\text{char}(K) \neq 2,3$ of picking a nonsingular semistable elliptic surface over $\mathbb{P}^{1}$ with two marked Weierstrass sections at $\infty$ and $0$ such that marked Weierstrass section at $0$ is 2-torsion or 3-torsion. In the end, we formulat…
Under some assumptions, we compute the Picard group of moduli of stable sheaves on Abelian surfaces. Considering relative moduli spaces, it is sufficient to consider the moduli of stable sheaves on the product of elliptic curves. By using elementary transformatios (which was used by Friedman to treat rank 2 moduli spaces on elliptic surfaces), we treat this case. We also show that some moduli spaces on P^2 are rational.
New local and global non-abelian zeta functions for elliptic curves are studied using certain refined Brill-Noether loci in moduli spaces of semi-stable bundles. Examples of these zeta functions and a justification of using only semi-stable bundles are given too. We end this paper with an appendix on the so-called Weierstrass Groups for general curves, which is motivated by a construction of Euler systems from torsion points (of elliptic curves).
In this paper, we study the moduli spaces $\mathcal{M}_{δ,c_2}$ of stable rank-2 vector bundles on non-K\" ahler elliptic surfaces, thus giving a classification these bundles; in the case of Hopf and Kodaira surfaces, these moduli spaces admit the structure of an algebraically completely integrable Hamiltonian system.
Assuming the finiteness of the Shafarevich-Tate group of elliptic curves over number fields we make several observations on the birational Grotendieck anabelian setion conjecture. We prove that the birational setion conjecture for curves over number fields can be reduced to the case of elliptic curves. In this case we prove that, as a consequence of a result of Stoll, a section of the exact sequence of the absolute Galois group of an elliptic curve over a number field arises from a rational point if and only if the induced section of the corresponding (geometrically abelianised) arithmetic fun…
U(1) symmetries play a central role in constructing phenomenologically viable F-theory compactifications that realize Grand Unified Theories (GUTs). In F-theory, gauge symmetries with abelian gauge factors are modeled by singular elliptic fibrations with additional rational sections, i.e. a non-trivial Mordell-Weil rank. To determine the full scope of possible low energy theories with abelian gauge factors, which allow for an F-theory realization, it is central to obtain a comprehensive list of all singular elliptic fibrations with extra sections. We answer this question for the case of one ab…
