Quadro-quadric special birational transformations from projective spaces to smooth complete intersections

TL;DR

This paper classifies specific quadro-quadric birational transformations between projective spaces and smooth complete intersections or hypersurfaces, focusing on cases where both the transformation and its inverse are defined by quadratic linear systems.

Contribution

It provides a classification of quadro-quadric special birational transformations with smooth base loci, extending understanding of their structure in algebraic geometry.

Findings

Classified transformations where Z is a complete intersection with smooth inverse base locus.

Classified transformations where Z is a hypersurface.

Identified conditions under which these transformations exist.

Abstract

Let \phi: \mathbb{P}^{r}\dashrightarrow Z be a birational transformation with a smooth connected base locus scheme, where Z\subseteq\mathbb{P}^{r+c} is a nondegenerate prime Fano manifold. We call \phi a quadro-quadric special briational transformation if \phi and \phi^{-1} are defined by linear subsystems of |\mathcal{O}_{\mathbb{P}^{r}}(2)| and |\mathcal{O}_{Z}(2)| respectively. In this paper we classify quadro-quadric special birational transformations in the cases where either (i) Z is a complete intersection and the base locus scheme of \phi^{-1} is smooth, or (ii) Z is a hypersurface.