We study automorphisms of order four on K3 surfaces. The symplectic ones have been first studied by Nikulin, they are known to fix six points and their action on the K3 lattice is unique. In this paper we give a classification of the purely non-symplectic automorphisms by relating the structure of their fixed locus to their action on cohomology, in the following cases: the fixed locus contains a curve of genus g>0; the fixed locus contains at least a curve and all the curves fixed by the square of the automorphism are rational. We give partial results in the other cases. Finally, we classify non-symplectic automorphisms of order four with symplectic square.
We present a method for computing all the symmetries of a rational ruled surface defined by a rational parametrization which works directly in parametric rational form, i.e. without computing or making use of the implicit equation of the surface. The method proceeds by translating the problem into the parameter space, and relies on polynomial system solving. If we want all the symmetries of the surface, including rotational symmetries, we need to deal with polynomial systems in four variables; if we are only interested in involutions (e.g. central symmetries, axial symmetries, reflections in a…
We complete the classification of order $5$ nonsymplectic automorphisms on hyper-Kähler fourfolds deformation equivalent to the Hilbert square of a K3 surface. We then compute the topological Lefschetz number of natural automorphisms of generalized Kummer fourfolds and we describe the geometry of their fix loci.
We provide explicit equations of some smooth complex quartic surfaces with many lines, including all 10 quartics with more than 52 lines. We study the relation between linear automorphisms and some configurations of lines such as twin lines and special lines. We answer a question by Oguiso on a determinantal presentation of the Fermat quartic surface.
In this paper we classify non-symplectic automorphisms of order 8 on complex K3 surfaces in case that the fourth power of the automorphism has only rational curves in its fixed locus. We show that the fixed locus is the disjoint union of a rational curve and 10 isolated points or it consists in 4 isolated fixed points. We give examples corresponding to the case with a rational curve in the fixed locus and to the case with only isolated points in the fixed locus.
We develop a characterization for the existence of symmetries of canal surfaces defined by a rational spine curve and rational radius function. In turn, this characterization inspires an algorithm for computing the symmetries of such canal surfaces. For Dupin cyclides in canonical form, we apply the characterization to derive an intrinsic description of their symmetries and symmetry groups, which gives rise to a method for computing the symmetries of a Dupin cyclide not necessarily in canonical form. As a final application, we discuss the construction of patches and blends of rational canal su…
We give a new approach to the symmetries of the Painlevé equations $P_{V},P_{IV},P_{III}$ and $P_{II}$, respectively. Moreover, we make natural extensions to fourth-order analogues for each of the Painlevé equations $P_{V}$ and $P_{III}$, respectively, which are natural in the sense that they preserve the symmetries.
In this paper, we propose a 4-parameter family of coupled Painlevé III systems in dimension four with affine Weyl group symmetry of type $D_4^{(1)}$. We also propose its symmetric form in which the $D_4^{(1)}$-symmetries become clearly visible.
In this paper, we propose a 3-parameter family of coupled Painlevé III systems in dimension four with affine Weyl group symmetry of type $A_5^{(2)}$. We also propose its symmetric form in which the $A_5^{(2)}$-symmetries become clearly visible.
Let $G\subset SO(4)$ denote a finite subgroup containing the Heisenberg group. In these notes we classify all these groups, we find the dimension of the spaces of $G$-invariant polynomials and we give equations for the generators whenever the space has dimension two. Then we complete the study of the corresponding $G$-invariant pencils of surfaces in $\mathbb{P}_3$ which we started in math. AG/0106080. It turns out that we have five more pencils, two of them containing surfaces with nodes.
We analyze freely-acting discrete symmetries of Calabi-Yau three-folds defined as hypersurfaces in ambient toric four-folds. An algorithm which allows the systematic classification of such symmetries which are linearly realised on the toric ambient space is devised. This algorithm is applied to all Calabi-Yau manifolds with $h^{1,1}(X)\leq 3$ obtained by triangulation from the Kreuzer-Skarke list, a list of some $350$ manifolds. All previously known freely-acting symmetries on these manifolds are correctly reproduced and we find five manifolds with freely-acting symmetries. These include a sin…
We introduce a characterization for affine equivalence of two surfaces of translation defined by either rational or meromorphic generators. In turn, this induces a similar characterization for minimal surfaces. In the rational case, our results provide algorithms for detecting affine equivalence of these surfaces, and therefore, in particular, the symmetries of a surface of translation or a minimal surface of the considered types. Additionally, we apply our results to designing surfaces of translation and minimal surfaces with symmetries, and to computing the symmetries of the higher-order Enn…
Let $W$ be a reflection group in a plane and $P$ a rational polygon that is invariant under the $W$-action. The action of $W$ on $P$ induces a $W$-action on the toric variety $X_P$ associated with $P$. In this paper, we study the $W$-representation on the cohomology $H^\ast(X_P)$ and show that the invariant subring $H^\ast(X_P)^W$ is isomorphic to the cohomology ring of the toric variety associated with the \emph{fundamental region}~$P/W$. As an example, we provide an explicit description of the main result for the case of the toric variety associated with the fan of Weyl chambers of type $G_2…
The automorphism group ${\rm Aut}\: X$ of a weighted homogeneous normal surface singularity $X$ has a maximal reductive algebraic subgroup $G$ which contains every reductive algebraic subgroup of ${\rm Aut}\: X$ up to conjugation. In all cases except the cyclic quotient singularities the connected component $G_1$ of the unit equals ${\Bbb C}^*$. The induced action of $G$ on the minimal good resolution of $X$ embeds the finite group $G/G_1$ into the automorphism group of the central curve $E_0$ of the exceptional divisor. We describe $G/G_1$ as a subgroup of ${\rm Aut}\: E_0$ in case $E_0$ is r…
Let $M$ be a smooth surface in $\mathbb R^3$ (or a complex surface in $\mathbb C^3$) and $k\geq 2$ be an integer. At any point on $M$ and for any plane in $\mathbb R^3$, we construct a holomorphic map-germ $(\mathbb C^2,0)\to(\mathbb C^3,0)$ of the form $F_k(x,y)= (x,y^k,f(x,y))$, called a $k$-folding map-germ. We study in this paper the local singularities of $k$-folding map-germs and relate them to the extrinsic differential geometry of $M$. More precisely, we (1) stratify the jet space of $k$-folding map-germs so that the strata of codimension $\le 4$ correspond to topologically equivalent …
We provide a method, based on Nikulin's lattice gluing techniques, which identifies the symplectic automorphisms of Kummer surfaces as permutation groups on 24 elements preserving the Golay code. In other words, we explicitly realise these symplectic automorphism groups as subgroups of the Mathieu group M_24. The example of the tetrahedral Kummer surface is treated in detail, confirming the existence proofs of Mukai and Kondo, that its group of symplectic automorphisms is a subgroup of one of eleven subgroups of the sporadic group known as Mathieu group M_23. Kondo's lattice construction, whic…
