Related papers: Spherical blow-ups of Grassmannians and Mori Dream…

Let $X$ be a smooth $n$-dimensional Fano hypersurface in $\mathbb P^{n+1}$ where $n \geq 3$. Let $\Gamma$ be a smooth positive-dimensional complete intersection of $X$, a hypersurface and one of more hyperplanes in $\mathbb P^{n+1}$. Let $Y…

We compute the facets of the effective cones of divisors on the blow-up of P^3 in up to five lines in general position. We prove that up to six lines these threefolds are weak Fano and hence Mori Dream Spaces.

We consider the problem to determine which blow-ups along subvarieties in products of two projective spaces are log Fano. By describing the nef cones of such blow-ups with special centers, we give a partial classification result. For each…

In this paper we study the pseudoeffective cones of blow-ups of Grassmannians at sets of points. For small numbers of points, the cones are often spanned by proper transforms of Schubert classes. In some special cases, we provide sharp…

We compute the cones of effective divisors on blowups of $\mathbb{P}^1 \times \mathbb{P}^2$ and $\mathbb{P}^1 \times \mathbb{P}^3$ in up to 6 points. We also show that all these varieties are log Fano, giving a conceptual explanation for…

We classify smooth toric Fano varieties of dimension $n\geq 3$ containing a toric divisor isomorphic to $\PP^{n-1}$. As a consequence of this classification, we show that any smooth complete toric variety $X$ of dimension $n\geq 3$ with a…

The goal of the present article is to survey the general theory of Mori Dream Spaces, with special regards to the question: When is the blow-up of toric variety at a general point a Mori Dream Space? We translate the question for toric…

We classify when the blowup of a complex Grassmannian $G(k, n)$ along a smooth Schubert subvariety $Z$ is Fano. We compute almost all the two-point, genus zero Gromov-Witten invariants of the blowup when $Z=G(k, n-1)$. We further prove a…

We consider blowups at a general point of weighted projective planes and, more generally, of toric surfaces with Picard number one. We give a unifying construction of negative curves on these blowups such that all previously known families…

We give a linear algebraic construction of the Lafforgue spaces associated to the Grassmannians $G(2,n)$ by blowing up certain explicitly defined monomial ideals, which sharpens and generalizes a result of Faltings. As an application, we…

We study blowups of weighted projective planes at a general point, and more generally blowups of toric surfaces of Picard number one. Based on the positive characteristic methods of Kurano and Nishida, we give a general method for…

We consider weak Fano manifolds with small contractions obtained by blowing up successively curves and subvarieties of codimension 2 in products of projective spaces. We give a classification result for a special case. In the process of…

In this paper we determine which blow-ups $X$ of $\mathbb{P}^n$ at general points are log Fano, that is, when there exists an effective $\mathbb{Q}$-divisor $\Delta$ such that $-(K_X+\Delta)$ is ample and the pair $(X,\Delta)$ is klt. For…

We explicitly construct the smooth toric Fano variety which is isomorphic to the blow-up of the projective space at torus invariant points in codimension one by anti-flips.

The effective cone of a Mori dream space admits two wall-and-chamber decompositions called Mori chamber and stable base locus decompositions. In general the former is a non trivial refinement of the latter. We investigate, from both the…

For every $n\geq 3$, we find a sufficient condition for the blow-up of a weighted projective space $\mathbb{P}(a,b,c,d_1,\cdots,d_{n-2})$ at the identity point not to be a Mori Dream Space. We exhibit several infinite sequences of weights…

Following Krah's method, we construct new examples of phantom categories as semiorthogonal components of the derived categories of two types of rational surfaces: the blowup of the plane at 11 points in general position, and the blowup of…

Fano varieties are subvarieties of the Grassmannian whose points parametrize linear subspaces contained in a given projective variety. These expository notes give an account of results on Fano varieties of complete intersections, with a…

The symmetric projective varieties of rank one are all smooth and Fano by a classic result of Akhiezer. We classify the locally factorial (respectively smooth) projective symmetric $G$-varieties of rank 2 which are Fano. When $G$ is…

We study the problem of counting pointed curves of fixed complex structure in blow-ups of projective space at general points. The geometric and virtual (Gromov-Witten) counts are found to agree asymptotically in the Fano (and some…