arXiv subjects
Publications and source records attributed to Stevell Muller.
In this short note we correct the lattice theoretic classification in [Rev. Mat. Iberoam. 38 (2022), no. 4, 1199--1218] of the first named author, including some missing cases and explaining how to recover them.
A projective hyperk\"ahler manifold of Kummer-type is said to be twisted modular if it is birational to the Albanese fiber of a moduli space of twisted sheaves on an abelian surface. We prove that, with the exception of certain cases of Picard rank 3, any projective Kummer-type manifold admitting a finite-order symplectic birational self-map that acts nontrivially on its second cohomology group is twisted modular. We provide a complete characterization of these exceptions in terms of their N\'eron-Severi lattices. We then investigate symplectic birational self-maps of modular Kummer-type manifolds, determining exactly which Mukai vectors allow the birational transformation induced by crossing the vertical wall, which acts on cohomology as a reflection, to correspond to a finite-order symplectic birational self-map. Additionally, we prove in an appendix several results concerning moduli spaces of twisted sheaves on abelian surfaces which were not readily available in the literature.
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.
Nikulin-type orbifolds are certain singular 4-dimensional irreducible holomorphic symplectic varieties. We show that the monodromy group of Nikulin-type orbifolds is maximal and classify finite order symplectic automorphisms up to deformation in terms of their action on the second integral cohomology group.
We extend the lattice-theoretic approach of Brandhorst--Cattaneo to classify algebraically trivial actions on the known IHS manifolds, up to deformation and birational conjugacy. In particular, we classify even order algebraically trivial nonsymplectic automorphisms, with or without trivial discriminant action. In the case of nontrivial discriminant actions, we show that such automorphisms exist only for finitely many known deformation types and orders.
We give a classification of finite groups of symplectic birational automorphisms on a manifold of K3^[3]-type with stable and stably saturated cohomological action. We describe the group of polarized automorphisms of a smooth double EPW-cube. Using this description, we exhibit examples of projective hyperkaehler manifolds of K3^[3]-type of maximal Picard rank with a symplectic action of a large group.
Given a complex vector space $V$ of finite dimension, its Grassmannian variety parametrizes all subspaces of $V$ of a given dimension. Similarly, if a finite group $G$ acts on $V$, its invariant Grassmannian parametrizes all the $G$-invariant subspaces of $V$ of a given dimension. Based on this fact, we develop an algorithm for computing $G$-invariant projective varieties arising as an intersection of hypersurfaces of the same degree. We apply the algorithm to find a projective model of a polarized K3 surface with a faithful action of $T_{192}\rtimes μ_2$ and some further symmetric K3 surfaces with a degree 8 polarization.
We classify finite groups that act faithfully by symplectic birational transformations on an irreducible holomorphic symplectic (IHS) manifold of OG10 type. In particular, if X is an IHS manifold of OG10 type and G a finite subgroup of symplectic birational transformations of X, then the action of G on H2(X, Z) is conjugate to a subgroup of one of 375 groups of isometries. We prove a criterion for when such a group is determined by a group of automorphisms acting on a cubic fourfold, and apply it to our classification. Our proof is computer aided and our results are available in a Zenodo dataset.
We give a complete classification of symplectic birational involutions of manifolds of $OG10$ type. We approach this classification with three techniques -- via involutions of the Leech lattice, via involutions of cubic fourfolds and finally lattice enumeration via a modified Kneser's neighbour algorithm. The classification consists of three involutions with an explicit geometric realisation via cubic fourfolds, and three exceptional involutions which cannot be obtained by any known construction.