Birational rigidity of orbifold degree 2 del Pezzo fibrations

TL;DR

This paper proves the birational rigidity of orbifold degree 2 del Pezzo fibrations over the projective line under certain conditions, extending known results from smooth cases to orbifold singularities.

Contribution

It establishes birational rigidity for orbifold degree 2 del Pezzo fibrations, generalizing previous results from smooth to orbifold cases with explicit conditions.

Findings

Orbifold degree 2 del Pezzo fibrations are birationally rigid under certain conditions.

They are not birational to Fano varieties, conic bundles, or other del Pezzo fibrations.

The results extend rigidity properties to orbifold singularities.

Abstract

Varieties fibered into del Pezzo surfaces form a class of possible outputs of the minimal model program. It is known that del Pezzo fibrations of degrees $1$ and $2$ over the projective line with smooth total space satisfying the so-called $K^2$-condition are birationally rigid: their Mori fibre space structure is unique. This implies that they are not birational to any Fano varieties, conic bundles or other del Pezzo fibrations. In particular, they are irrational. The families of del Pezzo fibrations with smooth total space of degree $2$ are rather special, as for "most" families a general del Pezzo fibration has the simplest orbifold singularities. We prove that orbifold del Pezzo fibrations of degree $2$ over the projective line satisfying explicit generality conditions as well as a generalised $K^2$-condition are birationally rigid.