arXiv subjects

Songbo Ling

Publications and source records attributed to Songbo Ling.

5 recordsLinked to original sources

arXiv2024

Albanese fibrations of surfaces with low slope

Let $S$ be a minimal irregular surface of general type, whose Albanese map induces a fibration $f:\,S \to C$ of genus $g$.We prove a linear upper bound on the genus $g$ if $K_S^2\leq 4\chi(\mathcal{O}_S)$. Examples are constructed showing that the above linear upper bound is sharp. We also give a characterization of the Albanese fibrations reaching the above upper bound when $\chi(\mathcal{O}_S)\geq 5$.On the other hand, we will construct a sequence of surfaces $S_n$ of general type with $K_{S_n}^2/\chi(\mathcal{O}_{S_n})>4$ and with an Albanese fibration $f_n$, such that the genus $g_n$ of a general fiber of $f_n$ increases quadratically with $\chi(\mathcal{O}_{S_n})$,and that $K_{S_n}^2/\chi(\mathcal{O}_{S_n})$ can be arbitrarily close to $4$.

math.AG↗

arXiv2024

Upper bounds on the genus of hyperelliptic Albanese fibrations

Let $S$ be a minimal irregular surface of general type, whose Albanese map induces a hyperelliptic fibration $f:\,S \to B$ of genus $g$.We prove a quadratic upper bound on the genus $g$, i.e., $g\leq h\big(\chi(\mathcal{O}_S)\big)$, where $h$ is a quadratic function. We also construct examples showing that the quadratic upper bounds can not be improved to the linear ones. In the special case when $p_g(S)=q(S)=1$, we show that $g\leq 14$.

math.AG↗

arXiv2018

Algebraic Surfaces with $p_g=q=1, K^2=4$ and Genus 3 Albanese Fibration

In this paper, we study the Gieseker moduli space $\mathcal{M}_{1,1}^{4,3}$ of minimal surfaces with $p_g=q=1, K^2=4$ and genus 3 Albanese fibration. Under the assumption that direct image of the canonical sheaf under the Albanese map is decomposable, we find two irreducible components of $\mathcal{M}_{1,1}^{4,3}$, one of dimension 5 and the other of dimension 4.

math.AG↗

arXiv2018

Algebraic Surfaces of General Type with $p_g=q=1$ and Genus 2 Albanese Fibrations

In this paper, we study algebraic surfaces of general type with $p_g=q=1$ and genus 2 Albanese fibrations. We first study the examples of surfaces with $p_g=q=1, K^2=5$ and genus 2 Albanese fibrations constructed by Catanese using singular bidouble covers of $\mathbb{P}^2$. We prove that these surfaces give an irreducible and connected component of $\mathcal{M}_{1,1}^{5,2}$, the Gieseker moduli space of surfaces of general type with $p_g=q=1, K^2=5$ and genus 2 Albanese fibrations. Then by constructing surfaces with $p_g=q=1,K^2=3$ and a genus 2 Albanese fibration such that the number of the summands of the direct image of the bicanonical sheaf (under the Albanese map) is 2, we give a negative answer to a question of Pignatelli.

math.AG↗

arXiv2017

Algebraic surfaces with $p_g=q=1, K^2=4$ and nonhyperelliptic Albanese fibrations of genus 4

In this paper we study minimal algebraic surfaces with $p_g=q=1,K^2=4$ and nonhyperelliptic Albanese fibrations of genus 4. We construct for the first time a family of such surfaces as complete intersections of type $(2,3)$ in a $\mathbb{P}^3$-bundle over an elliptic curve. For the surfaces we construct here, the direct image of the canonical sheaf under the Albanese map is decomposable (which is a topological invariant property). Moreover we prove that, all minimal surfaces with $p_g=q=1,K^2=4$ and nonhyperelliptic Albanese fibrations of genus 4 such that the direct image of the canonical sheaf under the Albanese map is decomposable are contained in our family. As a consequence, we show that these surfaces constitute a 4-dimensional irreducible subset $\mathcal{M}$ of $\mathcal{M}_{1,1}^{4,4}$, the Gieseker moduli space of minimal surfaces with $p_g=q=1, K^2=g=4$. Moreover, the closure of $\mathcal{M}$ is an irreducible component of $\mathcal{M}_{1,1}^{4,4}$.

math.AG↗