# Spherical blow-ups of Grassmannians and Mori Dream Spaces

## TL;DR

This paper classifies certain Fano varieties obtained by blowing up points, focusing on Grassmannians, and analyzes their geometric properties, including effective cones and stable base locus decompositions, establishing their Mori dream space status.

## Contribution

It provides a classification of spherical blow-ups of Grassmannians and computes their effective cones and stable base locus decompositions, highlighting their Mori dream space structure.

## Key findings

- Classified weak Fano varieties from blow-ups of prime Fano varieties.
- Computed effective cones of spherical blow-ups of Grassmannians.
- Determined stable base locus decomposition via hyperplanes and rational normal curves.

## 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.