Related papers: Hyperelliptic Szpiro inequality

Let $X$ be a semistable curve and $L$ a line bundle whose multidegree is uniform, i.e., in the range between those of the structure sheaf and the dualizing sheaf of $X$. We establish an upper bound for $h^0(X,L)$, which generalizes the…

We prove a new sharp asymptotic with the lower order term of zeroth order on $\mathcal{Z}_{\mathbb{F}_q(t)}(\mathcal{B})$ for counting the semistable elliptic curves over $\mathbb{F}_q(t)$ by the bounded height of discriminant $\Delta(X)$.…

We prove that any genus-2 Lefschetz fibration without reducible fibers and with ``transitive monodromy'' is holomorphic. The latter condition comprises all cases where the number of singular fibers is not congruent to 0 modulo 40. An…

We prove that every closed oriented smooth 4-manifold X admits a broken Lefschetz fibration (aka singular Lefschetz fibration) over the 2-sphere. Given any closed orientable surface F of square zero in X, we can choose the fibration so that…

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…

We study Lefschetz pencils on symplectic four-manifolds via the associated spheres in the moduli spaces of curves, and in particular their intersections with certain natural divisors. An invariant defined from such intersection numbers can…

Given a genus $g$ smooth Lefschetz fibration $\pi : M \to S^2$ with singular locus $\Delta \subseteq S^2$, we describe the subgroup $\operatorname{Br}(\pi)$ of the spherical braid group $\operatorname{Mod}(S^2,\Delta)$ consisting of braids…

For $g\geq 3$, we construct genus-$g$ Lefschetz fibrations over the two-sphere whose slopes are arbitrarily close to $2$. The total spaces of the Lefschetz fibrations can be chosen to be minimal and simply connected. It is also shown that…

According to Taubes, the Gromov invariants of a symplectic four-manifold X with b_+ > 1 satisfy the duality Gr(A) = +/- Gr(K-A), where K is Poincare dual to the canonical class. Extending joint work with Simon Donaldson in math.SG/0012067,…

This article presents the constructions of new infinite families of smooth 4-manifolds with the property that any two manifolds in the same family are homeomorphic and, from their construction, seem to be quite different, but cannot be…

In this paper we study a family of curves obtained by fibre products of hyperelliptic curves. We then exploit this family to construct examples of curves of given genus g over a finite field Fq with many rational points. The results…

In this paper, we shall prove Beauville's conjecture: if $f:S \to P^1$ is a non-trivial semistable fibration of genus g>1, then $f$ admits at least 5 singular fibers. We have also constructed an example of genus 2 with 5 singular fibers.…

The sum of Lyapunov exponents $L_f$ of a semi-stable fibration is the ratio of the degree of the Hodge bundle by the Euler characteristic of the base. This ratio is bounded from above by the Arakelov inequality. Sheng-Li Tan showed that for…

Let $(S,L)$ be a general polarized Enriques surface, with $L$ not numerically 2-divisible. We prove the existence of regular components of all Severi varieties of irreducible $\delta$-nodal curves in the linear system $|L|$, with $0\leq…

We analyze Higgs bundles $(V,\phi)$ on a class of elliptic surfaces $\pi:X\to B$, whose underlying vector bundle $V$ has vertical determinant and is fiberwise semistable. We prove that if the spectral curve of $V$ is reduced, then $\phi$ is…

We construct two types of non-holomorphic Lefschetz fibrations over $S^2$ with $(-1)$-sections ---hence, they are fiber sum indecomposable--- by giving the corresponding positive relators. One type of the two does not satisfy the slope…

We study a class of semistable ordinary hyperelliptic curves over 2-adic fields and the special fibre of their minimal regular model. We show that these curves can be controlled using `cluster pictures', similarly to the case of odd residue…

Let K be a field and let L/K be a finite extension. Let X/K be a scheme of finite type. A point of X(L) is said to be new if it does not belong to the union of X(F), when F runs over all proper subextensions of L. Fix now an integer g>0 and…

We define a new invariant $w$ for hyperelliptic Lefschetz fibrations over closed oriented surfaces, which counts the number of Dirac braids included intrinsically in the monodromy, by using chart description introduced by the second author.…

For each g > 2 and h > 1, we explicitly construct (1) fiber sum indecomposable relatively minimal genus g Lefschetz fibrations over genus h surfaces whose monodromies lie in the Torelli group, (2) fiber sum indecomposable genus g surface…