Related papers: On a smooth quartic surface containing 56 lines wh…

We prove that there exists a one-to-one correspondence between smooth quartic surfaces with an outer Galois point and K3 surfaces with a certain automorphism of order 4. Furthermore, we characterize quartic surfaces with two or more outer…

We give a complete equisingular deformation classification of simple spatial quartic surfaces which are in fact $K3$-surfaces.

An effective divisor D on a smooth (compact complex) surface X is called even, if its class $[D] \in H^2(X,\Z)$ is divisible by 2. D may be assumed reduced w.l.o.g. Then D being even is equivalent to the existence of a double cover $Y \to…

For any field k of characteristic at most 5 we exhibit an explicit smooth quartic surface in projective threespace over k with trivial automorphism group over the algebraic closure of k. We also show how this can be extended to higher…

We consider the action of the group $\mathrm{PGL}_4(K)$ on the smooth cubic surfaces of $\mathbb{P}^3_K$ ($K$ an algebraically closed field of characteristic zero). We classify, in an explicit way, all the smooth cubic surfaces with non…

In this, the first of three papers about $C_2$-equivariant complex quadrics, we calculate the equivariant ordinary cohomology of smooth antisymmetric quadrics. One of these quadrics coincides with a $C_2$-equivariant Grassmannian, and we…

In this, the last of three papers about $C_2$-equivariant complex quadrics, we complete the calculation of the equivariant ordinary cohomology of smooth symmetric quadrics in the cases where the fixed sets have more than two components.…

We construct a K3 surface over an algebraically closed field of characteristic 2 which contains two sets of 21 disjoint smooth rational curves such that each curve from one set intersects exactly 5 curves from the other set. This…

We show that there is a pair of smooth complex quartic K3 surfaces $S_1$ and $S_2$ in ${\mathbf P}^3$ such that $S_1$ and $S_2$ are isomorphic as abstract varieties but not Cremona isomorphic. We also show, in a geometrically explicit way,…

In [BN] the authors construct a special complex of degree 20 over M, which for an open three dimensional set parametrizes smooth complex surfaces of degree four invariant which are Heisenberg invariant and each member of the family contains…

For each integer $D\ge3$, we give a sharp bound on the number of lines contained in a smooth complex $2D$-polarized $K3$-surface in $\mathbb{P}^{D+1}$. In the two most interesting cases of sextics in $\mathbb{P}^4$ and octics in…

We construct an example of a smooth spatial quartic surface that contains 800 irreducible conics.

In 1884 the German mathematician Karl Rohn published a substantial paper on \cite{ROH} on the properties of quartic surfaces with triple points, proving (among many other things) that the maximum number of lines contained in a quartic…

We combine classical Vinberg's algorithms with the lattice-theoretic/arithmetic approach from arXiv:1706.05734 [math.AG] to give a method of classifying large line configurations on complex quasi-polarized K3-surfaces. We apply our method…

Consider a finite scheme of length l contained in a smooth quadric surface over the complex numbers. We determine the number of linearly independent curves passing through the scheme, of degree at least l - 1.

It is well-known that the Fermat surface of degree $d\geq 3$ has $3d^2$ lines. However, it has not yet been established what is the maximal number of pairwise disjoint lines that it can have if $d\geq 4$. In this article we show that the…

A recent result shows that a general smooth plane quartic can be recovered from its 24 inflection lines and a single inflection point. Nevertheless, the question whether or not a smooth plane curve of degree at least 4 is determined by its…

A smooth Hermitian surface $X$ is a projective surface isomorphic to the Fermat surface of degree $q+1$ in positive characteristic. We study incidence relations of the rational curves of degree $q+1$ contained in $X$, and show that such…

We study K3 surfaces of degree 6 containing two sets of 12 skew lines such that each line from a set intersects exactly six lines from the other set. These surfaces arise as hyperplane sections of the cubic line complex associated with the…

A smooth quartic curve in the complex projective plane has 36 inequivalent representations as a symmetric determinant of linear forms and 63 representations as a sum of three squares. These correspond to Cayley octads and Steiner complexes…