A type of the Lefschetz hyperplane section theorem on \Q-Fano 3-folds with Picard number one and $1/2(1,1,1)$-singularities

TL;DR

This paper proves a Lefschetz hyperplane section theorem for certain Q-Fano 3-folds with specific singularities, and constructs a new Calabi-Yau 3-fold with unique invariants using degeneration methods.

Contribution

It establishes a Lefschetz hyperplane section theorem for Q-Fano 3-folds with Picard number one and specific singularities, and provides a new example of a Calabi-Yau 3-fold with particular invariants.

Findings

Proved Lefschetz hyperplane section theorem for Q-Fano 3-folds with singularities.

Constructed a Calabi-Yau 3-fold with invariants (8, 44, -88).

Demonstrated the use of degeneration methods in this context.

Abstract

We prove a type of the Lefschetz hyperplane section theorem on Q-Fano 3-folds with Picard number one and $1/2(1,1,1)$-singularities by using some degeneration method. As a byproduct, we obtain a new example of a Calabi-Yau 3-fold $X$ with Picard number one whose invariants are $$(H_X^3, c_2 (X) \cdot H_X, {e} (X)) = (8, 44, -88),$$ where $H_X$, $e(X)$ and $c_2(X)$ are an ample generator of $\Pic(X)$, the topological Euler characteristic number and the second Chern class of $X$ respectively.