Related papers: Divisorial contractions to codimension three orbit…

We deal with a divisorial contraction in dimension 3 which contracts its exceptional divisor to a cA_1 point. We prove that any such contraction is obtained by a suitable weighted blow-up.

We deal with a divisorial contraction in dimension 3 which contracts its exceptional divisor to a smooth point. We prove that any such contraction can be obtained by a suitable weighted blow-up.

We prove that each divisorial contraction to a curve between terminal threefolds is a weighted blow-up under a suitable embedding. Moreover, we give a classification of the weighted blow-ups assuming that the curve is smooth.

We complete the explicit study of a three-fold divisorial contraction whose exceptional divisor contracts to a point, by treating the case where the point downstairs is a singularity of index $n \ge 2$. We prove that if this singularity is…

First, we simplify the existing classification due to Kawakita and Yamamoto of 3-dimensional divisorial contractions with centre a $cA_n$-singularity, also called compound $A_n$ singularity. Next, we describe the global algebraic divisorial…

Every three-fold divisorial contraction to a non-Gorenstein point is a weighted blow-up.

In this paper we discuss some recent results about extremal contractions of complex algebraic varieties. These are proper surjective maps, $\phi: X\longrightarrow Z$, of normal varieties with connected fibers such that $X$ has mild…

We prove that any three dimensional terminal singularity $P \in X$ can be resolved by a sequence of divisorial contractions with minimal discrepancies which are weighted blowups over points.

Let f: X -> Z be a local, projective, divisorial contraction between normal varieties of dimension n with Q-factorial singularities. Let $Y \subset X$ be a f-ample Cartier divisor and assume that f|Y: Y -> W has a structure of a weighted…

Let C be a smooth curve on an index 1 terminal 3-fold. We investigate the existence of extremal terminal divisorial contractions Y-->X that contract an irreducible surface E to C. We consider cases in respect to the singularities of the…

In this paper the three-dimensional divisorial contractions $f\colon Y\to (X\ni P)$ are classified provided that $\Exc f=E$ is an irreducible divisor, $f(E)=P$, the variety $Y$ has canonical singularities and $(X\ni P)$ is a toric terminal…

The purely log terminal blow-ups of three-dimensional terminal toric singularities are described. The three-dimensional divisorial contractions $f\colon (Y,E)\to (X\ni P)$ are described provided that $\Exc f=E$ is an irreducible divisor,…

We develop a ring-theoretic approach for blowing up many noncommutative projective surfaces. Let T be an elliptic algebra (meaning that, for some central element g of degree 1, T/gT is a twisted homogeneous coordinate ring of an elliptic…

We show that 3-fold terminal flips and divisorial contractions to a curve may be factored by a sequence of weighted blow-ups, flops, blow-downs to a locally complete intersection curve in a smooth 3-fold or divisorial contractions to a…

We consider $K_X$-negative extremal contractions $f\colon X\to (Z,o)$, where $X$ is an algebraic threefold with only $\epsilon$-log terminal Q-factorial singularities and $(Z,o)$ is a two (resp., one)-dimensional germ. The main result is…

We determine all the contractions within the class of finite-dimensional real Lie algebras whose coadjoint orbits have dimensions $\le2$.

We investigate Gromov-Witten invariants associated to exceptional classes for primitive birational contractions on a Calabi-Yau threefold X. It was observed in a previous paper that these invariants are locally defined, in that they can be…

Divisorial contractions to singularities, defined by equations $xy+z^n+u^n=0$ $n\ge3$ and $xy+z^3+u^4=0$ are classified.

Let $G$ be an algebraic group and let $X$ be a smooth $G$-variety with two orbits: an open orbit and a a closed orbit of codimension $1$. We give an algebraic description of the category of $G$-equivariant vector bundles on $X$ under a mild…

We show that under appropriate assumptions, a blown-up corona of a relatively hyperbolic group is equivariant and the compactification of the universal space for proper action by the blown-up corona is contractible. As a corollary, we…