On $K$-stability of $\mathbb{P}^3$ blown up along the disjoint union of a twisted cubic curve and a line

TL;DR

This paper classifies all K-polystable smooth Fano threefolds obtained by blowing up projective space along a disjoint union of a twisted cubic curve and a line, advancing understanding of stability conditions in algebraic geometry.

Contribution

It provides a complete classification of K-polystable Fano threefolds arising from specific blowup configurations, a previously unresolved problem.

Findings

Identifies all K-polystable cases for the given blowup configuration.

Establishes criteria for K-polystability in these threefolds.

Completes the classification of such Fano threefolds.

Abstract

We find all K-polystable smooth Fano threefolds that can be obtained as blowup of projective space along the disjoint union of a twisted cubic curve and a line.