arXiv subjects
Publications and source records attributed to Tiago Duarte Guerreiro.
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.
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 study the problem of determining when the blowup $X \to \mathbb{P}^3$ along a smooth space curve $C$ is a Mori Dream Space. We obtain sufficient conditions, as well obstructions to the Mori dreamness of $X$ based on the external geometry of $C$. We furthermore find infinitely many pairs $(g,d)$ such that the corresponding Hilbert schemes $H_{g,d}^S$ admit components whose general element has these obstructions. As a consequence we show that Mori dreamness is not an open property in flat families and exhibit various degenerational pathologies.
We survey what is known about Fano threefold weighted complete intersections from the point of view of birational rigidity.
We prove K-stability for infinitely many smooth members of the family 2.19 of the Mukai-Mori classification.
We introduce a new subclass of Fano varieties (Casagrande-Druel varieties), that are $n$-dimensional varieties constructed from Fano double covers of dimension $n-1$. We conjecture that a Casagrande-Druel variety is K-polystable if the double cover and its base space are K-polystable. We prove this for smoothable Casagrande-Druel threefolds, and for Casagrande-Druel varieties constructed from double covers of $\mathbb{P}^{n-1}$ ramified over smooth hypersurfaces of degree $2d$ with $n>d>\frac{n}{2}>1$. As an application, we describe the connected components of the K-moduli space parametrizing smoothable K-polystable Fano threefolds in the families 3.9 and 4.2 in the Mori-Mukai classification.
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 prove K-stability of every smooth member of the family 2.15 of the Mukai-Mori classification.
We complete the analysis on the birational rigidity of quasismooth Fano 3-fold deformation families appearing in the Graded Ring Database as a complete intersection. When such a deformation family $X$ has Fano index at least 2 and is minimally embedded in a weighted projective space in codimension 2, we determine which cyclic quotient singularity is a maximal centre. If a cyclic quotient singularity is a maximal centre, we construct a Sarkisov link to a non-isomorphic Mori fibre space or a birational involution. This allows, in particular, the construction of new examples of Fano 3-folds of codimension 6 which are realised as complete intersections in fake weighted projective spaces. We define linear cyclic quotient singularities on $X$ and prove that these are maximal centres by explicitly computing Sarkisov links centred at them. It turns out that each $X$ has a linear cyclic quotient singularity leading to a new birational model. As a consequence, we show that if $X$ is birationally rigid then its Fano index is 1. If the new birational model is a strict Mori fibre space, we determine its fibration type explicitly. In this case, a general member of $X$ is birational to a del Pezzo fibration of degrees 1, 2 or 3 or to a conic bundle $Y/S$ where $S$ is a weighted projective plane with at most $A_2$ singularities.
We provide a complete classification of paradoxical closed-loop $n$-linkages, where $n\geq6$, of mobility $n-4$ or higher, containing revolute, prismatic or helical joints. We also explicitly write down strong necessary conditions for $nR$-linkages of mobility $n-5$. Our main new tool is a geometric relation between a linkage $L$ and another linkage $L'$ resulting from adding equations to the configuration space of $L$. We then lift known classification results for $L'$ to $L$ using this relation.
We study Sarkisov links initiated by the toric weighted blowup of a point in $\mathbb{P}^3$ or $\mathbb{P}^4$ using variation of GIT. We completely classify which of these initiate Sarkisov links and describe the links explicitly. Moreover, if $X$ is the toric weighted blowup of $\mathbb{P}^d$ at a point, we give a simple criterion in terms of the weights of the blowup that characterises when $X$ is weak Fano.