Related papers: Special geometry of quartic curves

It is known for a long time that a nonsingular real algebraic curve of degree 2k in the projective plane cannot have more than 7/2*k^2-9/4*k+3/2$ even ovals. We show here that this upper bound is asymptotically sharp, that is to say we…

Motivated by strong desire to understand the natural geometry of moduli spaces of hyperbolic monopoles, we introduce and study a new type of geometry: pluricomplex geometry. It is a generalisation of hypercomplex geometry: we still have a…

A symmetry in quantum mechanics is described by the projective representations of a Lie symmetry group that transforms between physical quantum states such that the square of the modulus of the states is invariant. The Heisenberg…

We consider codimension 2 sphere congruences in pseudo-conformal geometry that are harmonic with respect to the conformal structure of an orthogonal surface. We characterise the orthogonal surfaces of such congruences as either $S$-Willmore…

We study homogeneous curves on some classes of reductive homogeneous spaces G=H which are geodesics with respect to any G-invariant metric on G=H. These curves are called equigeodesics. The spaces we consider are certain Stiefel manifolds…

Every action of a finite group scheme $G$ on a variety admits a projective equivariant model, but not necessarily a normal one. As a remedy, we introduce and explore the notion of $G$-normalization. In particular, every curve equipped with…

The paper is a contribution of the conjecture of Kobayashi that the complement of a generic plain curve of degree at least five is hyperbolic. The main result is that the complement of a generic configuration of three quadrics is hyperbolic…

An irreducible smooth projective curve over $\mathbb{F}\_q$ is called \textit{pointless} if it has no $\mathbb{F}\_q$-rational points. In this paper we study the lower existence bound on the genus of such a curve over a fixed finite field…

Two vertices $a$ and $b$ in a graph $X$ are cospectral if the vertex-deleted subgraphs $X\setminus a$ and $X\setminus b$ have the same characteristic polynomial. In this paper we investigate a strengthening of this relation on vertices,…

The 4-dimensional Sklyanin algebra is the homogeneous coordinate ring of a noncommutative analogue of projective 3-space. The degree-two component of the algebra contains a 2-dimensional subspace of central elements. The zero loci of those…

We present an efficient method for classifying the morphology of the intersection curve of two quadrics (QSIC) in PR3, 3D real projective space; here, the term morphology is used in a broad sense to mean the shape, topological, and…

This paper gives a complete answer of the following question: which (singular, projective) curves have a categorical resolution of singularities which admits a full exceptional collection? We prove that such full exceptional collection…

A very general surface of degree at least four in projective space of dimension three contains no curves other than intersections with surfaces. We find a formula for the degree of the locus of surfaces of degree at least five which contain…

We study the variety of rank $\leq k$ quadrics containing a general projective curve and show that it has the expected dimension in the range $g-d+r\leq 1$. By considering the loci where this expectation is not true, we construct new…

There are thirteen types of singular points for irreducible real quartic curves and seventeen types of singular points for reducible real quartic curves. This classification is originally due to D.A. Gudkov. There are nine types of singular…

In this paper we study genus-$4$ curves obtained as double covers of elliptic curves. Firstly we shall give explicit defining equations of such curves with explicit criterion for whether it is nonsingular, and show the irreducibility of the…

Let X be an arbitrary hyperbolic geodesic metric space and let G be a countable non-elementary weakly acylindrical group of isometries of X. We show that the second bounded cohomology group of G with real coefficients or with coefficients…

Let M_0^R be the moduli space of smooth real cubic surfaces. We show that each of its components admits a real hyperbolic structure. More precisely, one can remove some lower-dimensional geodesic subspaces from a real hyperbolic space H^4…

Uniformly quasiconformally homogeneous domains in $\mathbb{R}^n$ carry a transitive collection of $K$-quasiconformal maps for a fixed $K\geq 1.$ In this paper, we study two questions in this setting. The first is to show that…

Let k=F_q be a finite field of even characteristic. We obtain in this paper a complete classification, up to k-isomorphism, of non singular quartic plane curves defined over k. We find explicit rational normal models and we give closed…