arXiv subjects
Publications and source records attributed to Erik Paemurru.
This text is chapter 149 of the ZAG Handbook of Algebraic Geometry. We review the main results of the paper "Counting divisorial contractions with centre a $cA_n$-singularity", published in Publications of the Research Institute for Mathematical Sciences, Kyoto University.
We consider line bundles $\mathcal{O}(kd)$ on weighted projective spaces, where $k$ is an integer and $d$ is the least common multiple of the weights. Such line bundles are ample if and only if $k$ is positive. On the other hand, determining which line bundles are very ample is a delicate problem. We give various sharp criteria for very ampleness. As an example, if the weights are pairwise coprime, then $\mathcal{O}(d)$ is always very ample, which implies that general smooth well-formed weighted hypersurfaces of dimension at least two are simply connected. We also treat weighted blowups, relative very ampleness, projective normality, Rees rings and generation in degree 1 of Veronese subrings.
Let $X$ be a smooth $n$-dimensional Fano hypersurface in $\mathbb P^{n+1}$ where $n \geq 3$. Let $Γ$ be a smooth positive-dimensional complete intersection of $X$, a hypersurface and one of more hyperplanes in $\mathbb P^{n+1}$. Let $Y \to X$ be the blowup of $X$ along $Γ$. Let $φ\colon Y \rightarrow X$ be the blowup of $X$ along $Γ$. We describe the Mori chamber decomposition of $Y$ and its associated birational models. In particular, we show that $Y$ is a Mori dream space. We classify for which $X$ and $Γ$ the variety $Y$ is a Fano manifold and, if $X$ is a hyperplane, we classify the elementary Sarkisov links initiated by $φ$. Finally, we use this Mori chamber decomposition above to prove that certain Fano manifolds as above do not admit a Kähler-Einstein metric.
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.
We generalize an intersection-theoretic local inequality of Fulton-Lazarsfeld to weighted blowups. As a consequence, we obtain the $4n^2/(k+1)$-inequality for isolated $cA_k$ singularities, an analogue of the $4 n^2$-inequality for smooth points. We use this to prove birational rigidity of many families of Fano 3-fold weighted complete intersections with terminal quotient singularities and isolated $cA_k$ singularities, including sextic double solids with $cA_1$ and ordinary $cA_2$ points.
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 contractions corresponding to a given local analytic equivalence class of divisorial contractions with centre a point. Finally, 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.
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.
There is a proposition due to Kollár 1997 on computing log canonical thresholds of certain hypersurface germs using weighted blowups, which we extend to weighted blowups with non-negative weights. Using this, we show that the log canonical threshold of a convergent complex power series is at most $1/c$, where $(c, \ldots, c)$ is a point on a facet of its Newton polyhedron. Moreover, in the case $n = 2$, if the power series is weakly normalised with respect to this facet or the point $(c, c)$ belongs to two facets, then we have equality. This generalises a theorem of Varchenko 1982 to non-isolated singularities.
The $4 n^2$-inequality for smooth points plays an important role in the proofs of birational (super)rigidity. The main aim of this paper is to generalize such an inequality to terminal singular points of type $cA_1$, and obtain a $2 n^2$-inequality for $cA_1$ points. As applications, we prove birational (super)rigidity of sextic double solids, many other prime Fano 3-fold weighted complete intersections, and del Pezzo fibrations of degree $1$ over $\mathbb{P}^1$ satisfying the $K^2$-condition, all of which have at most terminal $cA_1$ singularities and terminal quotient singularities. These give first examples of birationally (super)rigid Fano 3-folds and del Pezzo fibrations admitting a $cA_1$ point which is not an ordinary double point.
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.