A remark on Ehresmann's Fibration Theorem

A remark on a question of Beauville about lagrangian fibrations

This note is a proof of the fact that a lagrangian torus on a hyperkaehler fourfold is always a fiber of an almost holomorphic lagrangian fibration.

Remarks on very basic slc-trivial fibrations

We study very basic slc-trivial fibrations. We show that restricting on any lc center of a very basic slc-trivial fibration, its moduli part is numerically trivial if and only if it is $\mathbb Q$-linearly trivial. We then prove that abundance conjecture for very basic slc-trivial fibrations holds true in dimension two when the moduli part is $\mathbb Q$-Cartier. As an application, we prove that the log canonical ring of a projective plt pair with Kodaira dimension 3 is finitely generated.

关于锥面纤维的理性和自映射群的评论

Remarks on the group of birational selfmaps of a conic fibration

研究锥面纤维的理性和自映射群，证明其存在到不可数多个$\mathbb Z/2\IZ$副本直接和的满射映射。

One remark on the b-semiampleness of the moduli part

In this short note we reduce the b-semiampleness conjecture for lc-trivial fibrations to the b-semiampleness conjecture for klt-trivial fibrations.

Some remarks on Brauer groups of K3 surfaces

We discuss the geometry of the genus one fibrations associated to an elliptic fibration on a K3 surface. We show that the two-torsion subgroup of the Brauer group of a general elliptic fibration is naturally isomorphic to the two-torsion of the Jacobian of a curve associated to the fibration. We remark that this is related to Recillas' trigonal construction. Finally we discuss the two-torsion in the Brauer group of a general K3 surface with a polarisation of degree two.

Remarks on Xiao's approach of Slope inequalities

We prove the slope inequality for a relative minimal surface fibration in positive characteristic via Xiao's approach. We also prove a better low bound for the slope of non-hyperelliptic fibrations.