Related papers: Self-dual polygons and self-dual curves

We investigate an `assumption of projectivity' that is appropriate to the self-dual axiomatic formulation of three-dimensional projective space.

An interesting class of automatic sequences emerges from iterated paperfolding. The sequences generate curves in the plane with an almost periodic structure. We generalize the results obtained by Davis and Knuth on the self-avoiding and…

We construct for any smooth projective curve of genus $q\ge 2$ with a fixed point free automorphism a nonisotrivial family of curves. Moreover we study the space of modular curves and that of parameters.

We apply contact homology to obtain new results in the problem of distinguishing immersed plane curves without dangerous self-tangencies.

We study rectangles inscribed in lines in the plane by parametrizing these rectangles in two ways, one involving slope and the other aspect ratio. This produces two paths, one that finds rectangles with specified slope and the other…

We study double line structures in projective spaces and quadric hypersurfaces, and investigate the geometry of irreducible components of Hilbert scheme of curves and moduli of stable sheaves of pure dimension 1 on a smooth quadric…

In this paper we solve the problem of analytic classification of plane curves singularities with two branches by presenting their normal forms. This is accomplished by means of a new analytic invariant that relates vectors in the tangent…

We classify plane curves $\mathcal{C}$ possessing two Galois points $P_1$ and $P_2 \in \mathbb{P}^2 \setminus \mathcal{C}$ such that the associated Galois groups $G_{P_1}$ and $G_{P_2}$ generate the semidirect product $G_{P_1}\rtimes…

We consider a plane polynomial vector field $P(x,y)dx+Q(x,y)dy$ of degree $m>1$. To each algebraic invariant curve of such a field we associate a compact Riemann surface with the meromorphic differential $\omega=dx/P=dy/Q$. The asymptotic…

We study lines through the origin of finite-dimensional complex vector spaces that enjoy a doubly transitive automorphism group. In doing so, we make fundamental connections with both discrete geometry and algebraic combinatorics. In…

We bound the genus of a projective curve lying on a complete intersection surface in terms of its degree and the degrees of the defining equations of the surface on which it lies.

A class of differential calculi is explored which is determined by a set of automorphisms of the underlying associative algebra. Several examples are presented. In particular, differential calculi on the quantum plane, the $h$-deformed…

A natural way to represent on the plane both a planar graph and its dual is to follow the definition of the dual, thus, to place vertices inside their corresponding primal faces, and to draw the dual edges so that they only cross their…

In this paper, we study the curvature properties of random complex plane curves. We bound from below the probability that a uniform proportion of the area of a random complex degree $d$ plane curve has a curvature smaller than $-d/8$. Our…

We prove the dynamical Mordell-Lang conjecture for product of endomorphisms of an affine curve and a projective curve over $\overline{\mathbb{Q}}$.

We study plane curves over finite fields whose tangent lines at smooth $\mathbb{F}_q$-points together cover all the points of $\mathbb{P}^2(\mathbb{F}_q)$.

We prove an unexpected general relation between the Jacobian syzygies of a projective hypersurface $V\subset \mathbb{P}^n$ with only isolated singularities and the nature of its singularities. This allows to establish a new method for the…

We develop a theory of simple pentagonal subdivision of quadrilateral tilings, on orientable as well as non-orientable surfaces. Then we apply the theory to answer questions related to pentagonal tilings of surfaces, especially those…

A bisection line divides a convex planar curve into two parts with equal areas. It is natural to study the envelope of these lines, which in general present singularities. The polygonal case is particularly inte\-resting, since there are…

We discuss various phenomena of tangency in projective and convex geometry.