Counting divisorial contractions with centre a cAncA_n-singularity

First, we simplify the existing classification due to Kawakita and Yamamoto of 3-dimensional divisorial contractions with centre a $cA_n$-singularity. Next, we consider divisorial contractions of discrepancy at least 2 to a fixed variety with centre a $cA_n$-singularity. We show that if there exists one such divisorial contraction, then there exist uncountably many such divisorial contractions.Comment: 16 page

Del Pezzo surfaces in weighted projective spaces

We study singular del Pezzo surfaces that are quasi-smooth and well-formed weighted hypersurfaces. We give an algorithm how to classify all of them

Log canonical thresholds of high multiplicity reduced plane curves

We compute log canonical thresholds of reduced plane curves of degree $d$ at points of multiplicity $d-1$. As a consequence, we describe all possible values of log canonical threshold that are less than $2/(d-1)$ for reduced plane curves of degree $d$. In addition, we compute log canonical thresholds for all reduced plane curves of degree less than 6.Comment: 13 page

Blowups of smooth Fano hypersurfaces, their birational geometry and divisorial stability

Let $\Gamma$ be a smooth $k$-dimensional hypersurface in $\mathbb P^{k+1}$ and $X \supset \Gamma$ a smooth $n$-dimensional Fano hypersurface in $\mathbb P^{n+1}$ where $n\geq 3$ and $k\geq 1$. Let $Y \rightarrow X$ be the blowup of $X$ along $\Gamma$. We give a constructive proof that $Y$ is a Mori dream space. In particular, we describe its Mori chamber decomposition and the associated birational models of $Y$. We classify for which $X$ and $\Gamma$ the variety $Y$ is a Fano manifold and we initiate the study of K-stability of $Y$.Comment: 20 page

Birational geometry of sextic double solids with a compound AnA_n singularity

Sextic double solids, double covers of $\mathbb P^3$ branched along a sextic surface, are the lowest degree Gorenstein Fano 3-folds, hence are expected to behave very rigidly in terms of birational geometry. Smooth sextic double solids, and those which are $\mathbb Q$-factorial with ordinary double points, are known to be birationally rigid. In this article, we study sextic double solids with an isolated compound $A_n$ singularity. We prove a sharp bound $n \leq 8$, describe models for each $n$ explicitly and prove that sextic double solids with $n > 3$ are birationally non-rigid

Birational models of terminal sextic double solids

We study sextic double solids, which are double covers of ℙ^3 branched along a sextic surface. We impose that such a 3-fold carries an isolated compound A_n singularity, abbreviated by cA_n. We first prove the bound n ≤ 8 for having an isolated cA_n singularity. We then explicitly parametrize sextic double solids with an isolated cA_n singularity and show that general sextic double solids with a cA_n point with n ≥ 4 are birationally non-rigid viewed as Mori fibre spaces. Birational non-rigidity is shown by constructing Sarkisov links starting with an extremal divisorial extraction from the singular point in each case