Global log canonical thresholds of minimal $(1,2)$-surfaces

TL;DR

This paper determines the lower bound of the global log canonical threshold for minimal (1,2)-surfaces and applies this to establish a volume inequality for certain threefolds, advancing understanding in algebraic geometry.

Contribution

It provides the first explicit lower bound for the global log canonical threshold of minimal (1,2)-surfaces and applies this to derive a new volume inequality for threefolds of general type.

Findings

Global log canonical threshold of minimal (1,2)-surfaces is at least 1/2.

Established a volume inequality for projective 3-folds with certain invariants.

Answered a specific open question related to Noether inequalities for threefolds.

Abstract

Let $S$ be a minimal surface of general type with $p_g(S)=2$ and $K^2_S=1$, so called by a minimal $(1,2)$-surface. Then we obtain that the global log canonical threshold of the surface $S$ via $K_S$ is greater than equal to $\frac{1}{2}$. As an application we have \[ {\rm{vol}}(X)\ge\frac{4}{3}p_g(X)-\frac{10}{3} \] for all projective $3$-folds $X$ of general type which answers Question 1.4 of [J. A. Chen, M. Chen, C. Jiang, "The Noether inequality for algebraic threefolds", arXiv:1803.05553] about Noether inequality for $X$ with $5\le p_g(X)\le 26$.