Birational transformations of threefold $\mathbf{Q}$-conic bundles

TL;DR

This paper develops an algorithm to convert threefold $Q$-conic bundles into a standard form, aiding classification and understanding of their birational properties in algebraic geometry.

Contribution

It introduces a systematic algorithm for transforming $Q$-conic bundles into their standard form, advancing the classification of these threefolds.

Findings

Algorithm successfully transforms $Q$-conic bundles to standard form.

Enhances understanding of birational transformations in algebraic geometry.

Facilitates classification of threefold $Q$-conic bundles.

Abstract

A $\mathbf{Q}$-conic bundle is a contraction $f: X\to Z$ of a three-dimensional algebraic variety $X$ to a surface~$Z$ such that the variety~$X$ has only terminal $\mathbf{Q}$-factorial singularities, the anticanonical divisor $-K_X$ is~$f$-ample, and $\uprho(X/Z)=1$. We provide an algorithm to transform a $\mathbf{Q}$-conic bundle to its standard form.