Nonbirational centers of linear projections of scrolls over curves (Q2105765)

Let \(\mathbf k\) be an algebraically closed field of characteristic zero. Let \(X \subset \mathbb P^N\) be a nondegenerate (i.e., not contained in any hyperplane) projective variety of dimension \(n\), codimension \(e\) and degree \(d\), hence, \(X\) is irreducible and reduced. We say that a point \(\in \mathbb P^N\) is a nonbirational center of \(X\) if the linear projection \(\pi_w : \mathbb P^N -\{w\} \rightarrow \mathbb P^{N-1}\) induces a nonbirational map of \(X\) to its image. By \(\mathcal B (X)\) we denote the set of all nonbirational centers out of \(X\), and by \(\mathcal C (X)\) we denote the set of all nonbirational centers in the smooth locus \(Sm X = X- Sing ( X )\) of \(X\). Then the closure of \(\mathcal B (X)\) and \(\mathcal C (X)\) are called the outer and the inner Segre locus of \(X\) respectively. Also \(\mathcal B (X)\) and \(\mathcal C (X)\) are called the strict outer and inner Segre locus of \(X\) respectively. We call that \(X \subset \mathbb P^N\) is a scroll over a curve if \(X\) is the image by a morphism of a projective bundle over a projective curve (i.e., variety of dimension 1) \(C\) such that the image of each fiber is linear in \(\mathbb P^N\). In this paper the author proves two results on the Segre locus for a scroll over a curve: First one concerns the emptiness of the Segre locus of an \(n\)-dimensional scroll over a curve which is not a cone with \(n-2\)-dimensional vertex and has no \(n-1\)-dimensional singular locus other than fibers; Second one gives some examples of scrolls with the nonempty Segre loci with some conditions. That is, the author shows that a scroll over a curve with some conditions has no nonbirational centers. Consequently such a nondegenerate scroll is cutting out by hypersurfaces of degree \(d - e + 1\) for its degree \(d\) and codimension \(e\) in the projective space. On the otherhand, examples of scrolls over curves with nonbirational centers are constructed.