Related papers: Noncommutative geometry of elliptic surfaces

Let $X$ be a quadratic vector field with a center whose generic orbits are algebraic curves of genus one. To each $X$ we associate an elliptic surface (a smooth complex compact surface which is a genus one fibration). We give the list of…

We study toroidal compactification of Matrix theory, using ideas and results of non-commutative geometry. We generalize this to compactification on the noncommutative torus, explain the classification of these backgrounds, and argue that…

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.…

Given a surface with boundary and some points on its boundary, a polygon diagram is a way to connect those points as vertices of non-overlapping polygons on the surface. Such polygon diagrams represent non-crossing permutations on a surface…

Let $C$ be a smooth projective curve defined over $\Qbar$, let $\pi:\mathcal{E}\lra C$ be an elliptic surface and let $\sigma_{P_1},\sigma_{P_2},\sigma_{Q}$ be sections of $\pi$ (corresponding to points $P_1,P_2, Q$ of the generic fiber $E$…

Examples of algebraic surfaces of general type with maximal Picard number are not abundant in the literature. Moreover, most known examples either possess low invariants, lie near the Noether line $K^2=2\chi-6$ or are somewhat scattered. A…

We describe in terms of the j-invariant all elliptic surfaces pi: X -> C with a section, such that h^{1,1}(X)=rank NS(X) and the Mordell-Weil group of pi is finite. We use this to give a complete solution to infinitesimal Torelli for…

In the past 15 years a study of ``noncommutative projective geometry'' has flourished. By using and generalizing techniques of commutative projective geometry, one can study certain noncommutative graded rings and obtain results for which…

This letter announces and summarizes results obtained in arXiv:1111.5051 and considers several natural extensions. The aforementioned paper proposes a procedure to reconstruct coefficients in a second-order, scalar, elliptic equation from…

We study elliptic surfaces over $\mathbb{Q}(T)$ with coefficients of a Weierstrass model being polynomials in $\mathbb{Q}[T]$ with degree at most 2. We derive an explicit expression for their rank over $\mathbb{Q}(T)$ depending on the…

We classify two dimensional integrable mappings by investigating the actions on the fiber space of rational elliptic surfaces. While the QRT mappings can be restricted on each fiber, there exist several classes of integrable mappings which…

We find new examples of complex surfaces with countably many non-isomorphic algebraic structures. Here is one such example: take an elliptic curve $E$ in $\mathbb P^2$ and blow up nine general points on $E$. Then the complement $M$ of the…

In this note, we initiate a study of the finite-dimensional representation theory of a class of algebras that correspond to noncommutative deformations of compact surfaces of arbitrary genus. Low dimensional representations are investigated…

In this note we study in detail the geometry of eight rational elliptic surfaces naturally associated to the sixteen reflexive polygons. The elliptic fibrations supported by these surfaces correspond under mirror symmetry to the eight…

The equivariant cohomology of certain moduli spaces of sheaves on isotrivial elliptic surfaces are shown to admit representations of infinite dimensional Lie (super)algebras. The construction is based on work of Billig and Chen-Li-Tan on…

This article describes recent applications of algebraic geometry to noncommutative algebra. These techniques have been particularly successful in describing graded algebras of small dimension.

We study the groups of automorphisms of rational algebraic surfaces that admit a relatively minimal pencil of curves of arithmetic genus one over an algebraically closed field of arbitrary characteristic. In particular, we classify such…

We determine the number of cusps of minimal Picard modular surfaces. The proof also counts cusps of other Picard modular surfaces of arithmetic interest. Consequently, for each N > 0 there are finitely many commensurability classes of…

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…

Given a conformally nonflat Einstein spacetime we define a fibration $P$ over it. The fibres of this fibration are elliptic curves (2-dimensional tori) or their degenerate counterparts. Their topology depends on the algebraic type of the…