Sarkisov Links with centres space curves on smooth cubic surfaces

TL;DR

This paper classifies Sarkisov links arising from blowing up smooth space curves on cubic surfaces, focusing on non-weak Fano cases, by analyzing multisecant curves and constructing links systematically.

Contribution

It provides a complete classification of such Sarkisov links for non-weak Fano cases, complementing previous work on weak Fano cases.

Findings

Classification of all relevant space curves on cubic surfaces.

Explicit construction of Sarkisov links for these curves.

Criteria for multisecant curves used in the classification.

Abstract

We construct and study Sarkisov links obtained by blowing up smooth space curves lying on smooth cubic surfaces. We restrict our attention to the case where the blowup is not weak Fano. Together with the results of arXiv:1106.3716 which cover the weak Fano case, we provide a classification of all such curves. This is achieved by computing all curves which satisfy certain necessary criteria on their multisecant curves and then constructing the Sarkisov link step by step.