Spherical blow-ups of Grassmannians and Mori dream spaces

Abstract

In this paper we classify weak Fano varieties that can be obtained by blowing-up general points in prime Fano varieties. We also classify spherical blow-ups of Grassmannians in general points, and we compute their effective cone. These blow-ups are, in particular, Mori dream spaces. Furthermore, we compute the stable base locus decomposition of the blow-up of a Grassmannian in one point, and we show how it is determined by linear systems of hyperplanes containing the osculating spaces of the Grassmannian at the blown-up point, and by the rational normal curves in the Grassmannian passing through the blown-up point.Throughout the paper X will be a normal projective variety over an algebraically closed field of characteristic zero. We denote by N 1 (X) the real vector space of R-Cartier divisors modulo numerical equivalence. The nef cone of X is the closed convex cone Nef(X) ⊂ N 1 (X) generated by classes of nef divisors. The movable cone of X is the convex cone Mov(X) ⊂ N 1 (X) generated by

Ample bodies and Terracini loci of projective varieties

“…Proof By [13, Lemma 6.5], we have the following characterization of the tangent space of the Grassmannian at a point $$[U]\in \mathbb {G}(r,n)$$: $$$$\begin{equation*} T_{[U]}(\mathbb {G}(r,n)) =\langle e_I \:|\: d(I,\lbrace 0,\dots ,r\rbrace )\leqslant 1 \rangle , \end{equation*}$$$$where $$(e_0,\dots ,e_n)$$ is a basis of $$k^{n+1}$$, $$U$$ is generated by $$e_0,\dots ,e_r\in \mathbb {P}^n$$, $$e_I = e_{i_0}\wedge \cdots \wedge e_{i_r}$$ and $$d(I,J)$$ is the Hamming distance between the two lists $$I$$ and $$J$$. Now, given $$[U],[V]\in \mathbb {...$…”

On Mori chamber and stable base locus decompositions

“…Let Gpr, nq be the Grassmannian parametrizing r-planes in P n , and Gpr, nq k the blow-up of P n at k general points. These blow-ups have been studied in [MR18]. In particular the stable base locus decomposition of EffpGpr, nq 1 q has been computed in [MR18,Theorem 1.3].…”

“…These blow-ups have been studied in [MR18]. In particular the stable base locus decomposition of EffpGpr, nq 1 q has been computed in [MR18,Theorem 1.3].…”