Related papers: Plane sextics with a type $\mathbf{E}_6$ singular …
Given a birational normal extension S of a two-dimensional local regular ring R, we describe all the equisingularity types of the complete ideals J in R whose blowing-up has some point at which the local ring is analytically isomorphic to…
An EPW-sextic is a special 4-dimensional hypersurfaces of degree 6 which comes equipped with a double cover which generically is a Hyperkaehler 4-fold deformation equivalent to the Hilbert square of a K3 surface. The family of EPW-sextics…
We upgrade the classical operation of \textit{isomonodromic deformations} along a path $\gamma$ to a functor $\mathbb{P}_{\gamma}$ between categories of flat connections with logarithmic singularities along a divisor $D$, which itself…
We characterise, in terms of Dixmier-Ohno invariants, the types of singularities that a plane quartic curve can have. We then use these results to obtain new criteria for determining the stable reduction types of non-hyperelliptic curves of…
O'Grady showed that certain special sextics in $\mathbb{P}^5$ called EPW sextics admit smooth double covers with a holomorphic symplectic structure. We propose another perspective on these symplectic manifolds, by showing that they can be…
Following our previous work, we develop an algorithm to compute a presentation of the fundamental group of certain partial compactifications of the complement of a complex arrangement of lines in the projective plane. It applies, in…
In this paper we treat the intersection of fixed point subgroups by the involutive automorphisms of exceptional Lie group $G= F_4, E_6, E_7$. We shall find involutive automorphisms of $G$ such that the connected component of the…
For every octonion division algebra O, there exists a projective plane which is parametrized by O; these planes are related to rank two forms of linear algebraic groups of absolute type E6. We study all possible polarities of such octonion…
Alexander polynomials of sextics with only simple singularities or sextics of torus type with arbitrary singularities are computed. We show that for ieeducible sextics,there are four possibilities: $(t^2-t+1)^j, j=0,1,2,3$.
We study families of plane algebraic curves sharing the same set of foci. We reformulate confocality via a focal map on equiclassical families and analyze its fibers using deformation theory.
We study hyperplane sections of smooth polarized $K3$-surfaces that split into unions of lines. We describe the dual adjacency graphs of such sections and find sharp upper bounds on their number. In most cases (starting from degree $6$), we…
Lecture 1: Projective and K\"ahler Manifolds, the Enriques classification, construction techniques. Lecture 2: Surfaces of general type and their Canonical models. Deformation equivalence and singularities. Lecture 3: Deformation and…
Consider equilateral pentagons $V_1,\ldots,V_5$ in the Euclidean plane. When we identify pentagons that differ by translation, rotation, and magnification, the moduli space of possible shapes that we get is an oft-studied polygon space: a…
We study the symmetry groups and winding numbers of planar curves obtained as images of weighted sums of exponentials. More generally, we study the image of the complex unit circle under a finite or infinite Laurent series using a…
We show that a homology plane of general type has at worst a single cyclic quotient singular point. An example of such a surface with a singular point does exist. We also show that the automorphism group of a smooth contractible surface of…
We enumerate complex algebraic hypersurfaces in $P^n$, of a given (high) degree with one singular point of a given singularity type. Our approach is to compute the (co)homology classes of the corresponding equi-singular strata in the…
For $\mathrm{O}(\mathrm{q},k)$, the orthogonal group over a field $k$ of characteristic 2 with respect to a quadratic form $\mathrm{q}$, we discuss the isomorphism classes of fixed points of involutions. When the quadratic space is either…
In this work we present a useful way to introduce the octonionic projective and hyperbolic plane through the use of Veronese vectors. Then we focus on their relation with the exceptional Jordan algebra and show that the Veronese vectors are…
We give a new computer-assisted proof of the classification of maximal subgroups of the simple group ${}^2E_6(2)$ and its extensions by any subgroup of the outer automorphism group $S_3$. This is not a new result, but no earlier proof…
Pfister and Steenbrink studied punctual Hilbert schemes for irreducible curve singularities. In particular, they investigated the structure of special punctual Hilbert schemes for certain monomial curve singularities. In this paper, we…