Varieties of minimal rational tangents of codimension 1 (Q2854296)

There has been lots of work on characterizing projective varieties by special properties of rational curves on them after the pioneering work of Mori's solution of the Hartshorne conjecture. The paper under review is an attempt along these lines.NEWLINENEWLINEGiven a smooth projective uniruled variety \(X\) and a general point \(x\), there exists a proper family \(\mathcal{K}\) of smooth rational curves, called the minimal rational curves. Define \(\deg \mathcal{K}\) to be the intersection number of \(-K_X\) with a rational curve in this family. The paper describes the structure of the variety \(X\) when \(\deg \mathcal{K}=\dim X\) under some conditions.NEWLINENEWLINEThe first main result is Theorem 1.4, which shows that such varieties (with some conditions satified) are birationally quotients of the blow-up of \(\mathbb{P}^n\) along the intersection of a codimension one linear subspace and a smooth hypersurface by a finite group.NEWLINENEWLINEThe proof is technical, and based on a study of the variety of minimal rational tangenet (VMRT) coming from the family \(\mathcal{K}\). The author also proves some technical results on when the VMRT-structure is locally flat, namely, Theorem 1.9 and 1.10. We refer to Definition 3.14 in the paper for locally flatness of the VMRT-structure.NEWLINENEWLINEAs an application of the technical results, the author gives a characterization of the hyperquadric in terms of the family of minimal rational curves (Theorem 1.11).