arXiv subjects
Publications and source records attributed to Livia Campo.
We prove the K-polystability of the general Fano complete intersection of arbitrary multidegree and dimension, and the K-stability of the general Fano complete intersection that is not isomorphic to projective space or a quadric hypersurface. We prove analogous results for certain smooth weighted complete intersections.
We develop a framework to study the K-stability of weighted Fano hypersurfaces based on a combination of birational and convex-geometric techniques. As an application, we prove that all quasi-smooth weighted Fano hypersurfaces of index 1 with at most two weights greater than 1 are K-stable. We also construct several examples of K-unstable quasi-smooth weighted Fano hypersurfaces of low indices. To prove these results, we establish lower bounds for stability thresholds using the method of Abban-Zhuang, which reduces the problem to lower-dimensional cases. A key feature of our approach is the use of plt flags that are not necessarily admissible.
We settle the problem of K-stability of quasi-smooth Fano 3-fold hypersurfaces with Fano index 1 by providing lower bounds for their delta invariants. We use the method introduced by Abban and Zhuang for computing lower bounds of delta invariants on flags of hypersurfaces in the Fano 3-fold.
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 $Y \to X$ be the blowup of $X$ along $\Gamma$. Let $\varphi \colon Y \rightarrow X$ be the blowup of $X$ along $\Gamma$. 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 $\Gamma$ the variety $Y$ is a Fano manifold and, if $X$ is a hyperplane, we classify the elementary Sarkisov links initiated by $\varphi$. Finally, we use this Mori chamber decomposition above to prove that certain Fano manifolds as above do not admit a K\"ahler-Einstein metric.
We give conditions for a uniruled variety of dimension at least 2 to be non-solid. This study provides further evidence to a conjecture by Abban and Okada on the solidity of Fano 3-folds. To complement our results we write explicit birational links from Fano 3-folds of high codimension embedded in weighted projective spaces.
We show that five of Reid's Fano 3-fold hyperurfaces containing at least one compound Du Val singularity of type $cA_n$ have pliability at least two. The two elements of the pliability set are the singular hypersurface itself, and another non-isomorphic Fano hypersurface of the same degree, embedded in the same weighted projective space, but with different compound Du Val singularities. The birational map between them is the composition of two birational links initiated by blowing up two Type I centres on a codimension 4 Fano 3-fold of $\mathbb{P}^2 \times \mathbb{P}^2$-type having Picard rank 2.
We construct a deformation family for each of the 34 Hilbert series of index 2 Fano 3-folds. In 18 cases we construct two different families, distinguished by the topology of their general members.
We classify Sarkisov links from index 1 Fano 3-folds anticanonically embedded in codimension 4 that start from so-called Type I Tom centres. We apply this to compute the Picard rank of many such Fano 3-folds.