We give a systematic method to calculate some homological data from the global monodromy of a topological elliptic surface. We apply this method to the cases 1) the transcendental lattice of an extremal elliptic K3 surface, 2) the torsion part of Mordell-Weil group of a general elliptic surface, and 3) the Mordell-Weil lattice of a rational elliptic surface. We present possible list of global monodromies of rational elliptic surface and extremal elliptic $K3$ surface which reproduce to the classification obtained by Oguiso-Shioda and Shimada-Zhang respectively.
We discuss the equivalence between the categories of certain ribbon graphs and subgroups of the modular group $Γ$ and use it to construct exponentially large families of not Hurwitz equivalent simple braid monodromy factorizations of the same element. As an application, we also obtain exponentially large families of {\it topologically} distinct algebraic objects such as extremal elliptic surfaces, real trigonal curves, and real elliptic surfaces.
Towards the study of the representation theory of any dihedral Artin group B, we build rational morphisms from B to the group of invertible elements of the associated infinitesimal braids algebra. For this we build analogues of Drinfeld associators, in rank 2 for arbitrary dihedral symmetries of the plane.
In \cite{K-rig}, a map $β:\mathcal R\to\mathcal{B}el$ from the set $\mathcal R$ of equivalence classes of rigid germs of finite morphisms branched in germs of curves having $ADE$ singularity types onto the set $\mathcal{B}el$ of rational Belyi pairs $f:\mathbb P^1\to\mathbb P^1$ considered up to the action of $PGL(2,\mathbb C)$ was defined. In this article, the inverse images of this map are investigated in terms of monodromies of Belyi pairs.
In this paper we determine a minimal set of generators for the Cox rings of extremal rational elliptic surfaces. Moreover, we develop a technique for computing the ideal of relations between them which allows, in all but three cases, to provide a presentation of the Cox ring.
Miranda and Persson classified all extremal rational elliptic surfaces in characteristic zero. We show that each surface in Miranda and Persson's classification has an integral model with good reduction everywhere (except for those of type X_{11}(j), which is an exceptional case), and that every extremal rational elliptic surface over an algebraically closed field of characteristic p > 0 can be obtained by reducing one of these integral models mod p.
We review and develop some techniques used to investigate the effective cones of higher codimension classes. Our results show that a large collection of boundary strata of rational tails type are extremal in their effective cones on $\overline{\mathcal{M}}_{g,n}$ and provide evidence for the conjecture that all boundary strata of $\overline{\mathcal{M}}_{g,n}$ are extremal. As a corollary, we show that all boundary strata are extremal in genus zero.
We consider a rational surface with a relatively minimal fibration. Picard number of a such fibred surface is bounded in terms of the genus of a general fibre. When Picard number is the maximum for any given genus, we characterize a such fibred surface whose Mordell-Weil group is trivial by singular fibres. Furthermore, we describe the defining equation explicitly.
We study K3 surfaces over a number field $k$ which are double covers of extremal rational elliptic surfaces. We provide a list of all elliptic fibrations on certain K3 surfaces together with the degree of a field extension over which each genus one fibration is defined and admits a section. We show that the latter depends, in general, on the action of the cover involution on the fibers of the genus 1 fibration.
We prove some sufficient conditions in order that a root of the Bernstein-Sato polynomial contributes to a difference between certain D-modules generated by rational powers of a holomorphic function; for instance, this holds in the case of isolated singularities with semisimple Milnor monodromies. We then construct an example where a root does not contribute to a difference. This also solves an old open problem about the relation between the Milnor monodromy and the exponential of the residue of the Gauss-Manin connection on the saturation of the Brieskorn lattice. This shows that the structur…
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…
We study effective divisors on $\overline{M}_{0,n}$, focusing on hypertree divisors introduced by Castravet and Tevelev and the proper transforms of divisors on $\overline{M}_{1,n-2}$ introduced by Chen and Coskun. Results include a database of hypertree divisor classes and closed formulas for Chen--Coskun divisor classes. We relate these two types of divisors, and from this construct extremal divisors on $\overline{M}_{0,n}$ for $n \geq 7$ that furnish counterexamples to the conjectural description of the effective cone of $\overline{M}_{0,n}$ given by Castravet and Tevelev.
In view of Mori theory, rational homogenous manifolds satisfy a recursive condition: every elementary contraction is a rational homogeneous fibration and the image of any elementary contraction also satisfies the same property. In this paper, we show that a smooth Fano $n$-fold with the same condition and Picard number greater than $n-6$ is either a rational homogeneous manifold or the product of $n-7$ copies of $\mathbb{P}^1$ and a Fano $7$-fold $X_0$ constructed by G. Ottaviani. We also clarify that $X_0$ has non-nef tangent bundle and in particular is not rational homogeneous.
A new approach to the analytic theory of difference equations with rational and elliptic coefficients is proposed. It is based on the construction of canonical meromorphic solutions which are analytical along "thick paths". The concept of such solutions leads to a notion of local monodromies of difference equations. It is shown that in the continuous limit they converge to the monodromy matrices of differential equations. New type of isomonodromic deformations of difference equations with elliptic coefficients changing the periods of elliptic curves is constructed.
The minimum of intersection numbers of the anti-canonical divisor with rational curves on a Fano manifold is called pseudo-index. It is expected that the intersection number of anti-canonical divisor attains to the minimum on an extremal ray, i.e. there exists an extremal rational curve whose intersection number with the anti-canonical divisor equals the pseudo-index. In this note, we prove this for smooth Fano 4-folds having birational contractions.
We study resonant billiard trajectories within quadrics in the $d$-dimensional Euclidean space. We relate them to the theory of approximation, in particular the extremal rational functions on the systems of $d$ intervals on the real line. This fruitful link enables us to prove fundamental properties of the billiard dynamics and to provide a comprehensive study of a large class of non-periodic trajectories of integrable billiards. A key ingredient is a functional-polynomial relation of a generalized Pell type. Applying further these ideas and techniques to $s$-weak billiard trajectories, we com…
In the present work we study parameter spaces of two line point configurations introduced by Böröczky. These configurations are extremal from the point of view of Dirac-Motzkin Conjecture settled recently by Green and Tao. They have appeared also recently in commutative algebra in connection with the containment problem for symbolic and ordinary powers of homogeneous ideals and in algebraic geometry in considerations revolving around the Bounded Negativity Conjecture. Our main results are Theorem A and Theorem B. We show that the parameter space of what we call $B12$ configurations is a three …
We investigate configurations of rational double points with the total Milnor number 21 on supersingular $K3$ surfaces. The complete list of possible configurations is given. As an application, we also give the complete list of extremal (quasi-)elliptic fibrations on supersingular $K3$ surfaces.
Let (C,0) be a reduced curve germ in a normal surface singularity (X,0). The main goal is to recover the delta invariant of the abstract curve (C,0) from the topology of the embedding. We give explicit formulae whenever (C,0) is minimal generic and (X,0) is rational (as a continuation of previous works of the authors). Additionally we prove that if (X,0) is a quotient singularity, then the delta invariant of (C,0) only admits the values r-1 or r, where r is the number or irreducible components of (C,0). (r-1 realizes the extremal lower bound, valid only for `ordinary r-tuples'.)
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.
Given an algebraic surface $X$, the Hilbert scheme $X^{[n]}$ of $n$-points on $X$ admits a contraction morphism to the $n$-fold symmetric product $X^{(n)}$ with the extremal ray generated by a class $β_n$ of a rational curve. We determine the two point extremal GW-invariants of $X^{[n]}$ with respect to the class $dβ_n$ for a simply-connected projective surface $X$ and the quantum first Chern class operator of the tautological bundle on $X^{[n]}$. The methods used are vertex algebraic description of $H^*(X^{[n]})$, the localization technique applied to $X=\mathbb P^2$, and a generalization of …
We classify minimal pairs (X, G) for smooth rational projective surface X and finite group G of automorphisms on X. We also determine the fixed locus X^G and the quotient surface Y = X/G as well as the fundamental group of the smooth part of Y. The realization of each pair is included. Mori's extremal ray theory and recent results of Alexeev and also Ambro on the existence of good anti-canonical divisors are used.
We provide an explicit resolution of the existence problem for extremal Kaehler metrics on toric 4-orbifolds M with second Betti number b2(M)=2. More precisely we show that M admits such a metric if and only if its rational Delzant polytope (which is a labelled quadrilateral) is K-polystable in the relative, toric sense (as studied by S. Donaldson, E. Legendre, G. Szekelyhidi et al.). Furthermore, in this case, the extremal Kaehler metric is ambitoric, i.e., compatible with a conformally equivalent, oppositely oriented toric Kaehler metric, which turns out to be extremal as well. These results…
We study the geometry of the smooth projective surfaces that are defined by Frobenius forms, a class of homogenous polynomials in prime characteristic recently shown to have minimal possible F-pure threshold among forms of the same degree. We call these surfaces $\textit{extremal surfaces}$, and show that their geometry is reminiscent of the geometry of smooth cubic surfaces, especially non-Frobenius split cubic surfaces of characteristic two, which are examples of extremal surfaces. For example, we show that an extremal surface $X$ contains $d^2(d^2-3d+3)$ lines where $d$ is the degree, which…
Let X be a K3 surface with an involution g which has non-empty fixed locus X^g and acts non-trivially on a non-zero holomorphic 2-form. We shall construct all such pairs (X, g) in a canonical way, from some better known double coverings of log del Pezzo surfaces of index at most 2 or rational elliptic surfaces, and construct the only family of each of the three extremal cases where X^g contains 10 (maximum possible) curves. We also classify rational log Enriques surfaces of index 2. Our approach is more geometrical rather than lattice-theoretical (see Nikulin's paper for the latter approach).
We derive formulas for the construction of all inequivalent Jacobian elliptic fibrations on the Kummer surface of two non-isogeneous elliptic curves from extremal rational elliptic surfaces by rational base transformations and quadratic twists. We then show that each such decomposition yields a description of the Picard-Fuchs system satisfied by the periods of the holomorphic two-form as either a tensor product of two Gauss' hypergeometric differential equations, an Appell hypergeometric system, or a GKZ differential system. As the answer must be independent of the fibration used, identities r…
We study F-theory compactifications with U(1)xU(1) gauge symmetry on elliptically fibered Calabi-Yau manifolds with a rank two Mordell-Weil group. We find that the natural presentation of an elliptic curve E with two rational points and a zero point is the generic Calabi-Yau onefold in dP_2. We determine the birational map to its Tate and Weierstrass form and the coordinates of the two rational points in Weierstrass form. We discuss its resolved elliptic fibrations over a general base B and classify them in the case of B=P^2. A thorough analysis of the generic codimension two singularities of …
