Related papers: Tritangents and Their Space Sextics
Our main result is an effective version of the Torelli theorem in genus $3$ and any characteristic not $2$: the configuration of the odd theta characteristics of a curve $C$ of genus $3$ determines a del Pezzo surface $S$ of degree two and…
The paper is an introduction to the use of the classical Newton-Puiseux procedure, oriented to an algorithmic description of it. This procedure enables to get polynomial approximations for parameterizations of branches of an algebraic plane…
We use a probabilistic interpretation of solid angles to generalize the well-known fact that the inner angles of a triangle sum to 180 degrees. For the 3-dimensional case, we show that the sum of the solid inner vertex angles of a…
Two Magma functions are given: one computes linear systems of plane curves with non-ordinary singularities and the other computes a scheme which parametrizes given degree plane curves with given singularities. These functions provide an…
Let $C$ be a curve of genus two. We denote by $SU_C(3)$ the moduli space of semi-stable vector bundles of rank 3 and trivial determinant over $C$, and by $J^d$ the variety of line bundles of degree $d$ on $C$. In particular, $J^1$ has a…
A new formula is obtained in algebraic topology, in terms of Betti numbers, and a new method, called the spinal method, is suggested and developed for generating quadrangulations of closed orientable surfaces. Those surfaces arise as the…
We investigate plane curves intersecting in at most two unibranched points to study the algebraic exceptional set appearing in standard conjectures of diophantine and hyperbolic geometry. Our first result compares the local geometry of two…
This paper provides an in-depth analysis of how computational algebraic geometry can be used to deal with the problem of counting and classifying $r\times s$ partial Latin rectangles based on $n$ symbols of a given size, shape, type or…
A tiling of the sphere by triangles, squares, or hexagons is convex if every vertex has at most 6, 4, or 3 polygons adjacent to it, respectively. Assigning an appropriate weight to any tiling, our main result is explicit formulas for the…
We wish to draw attention to an interesting and promising interaction of two theories. On the one hand, it is the theory of \textbf{pseudo-triangulations} which was useful for implicit solution of thecarpenter's rule problem and proved…
Each elliptic curve can be embedded uniquely in the projective plane, up to projective equivalence. The hessian curve of the embedding is generically a new elliptic curve, whose isomorphism type depends only on that of the initial elliptic…
The computation of the dimension of linear systems of plane curves through a bunch of given multiple points is one of the most classic issues in Algebraic Geometry. In general, it is still an open problem to understand when the points fail…
Let S be a surface in complex projective 3-space, having only nodes as singularities. Suppose that S has degree 6. We show that the maximum number of nodes which S can have is 65. An abbreviated history of this is as follows. Basset showed…
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…
A new method to obtain trigonometry for the real spaces of constant curvature and metric of any (even degenerate) signature is presented. The method encapsulates trigonometry for all these spaces into a single basic trigonometric group…
It is shown that there exists a twistor space on the $n$-fold connected sum of complex projective planes $n\mathbb{CP}^2$, whose algebraic dimension is one and whose general fiber of the algebraic reduction is birational to an elliptic…
Indirect hex-dominant meshing methods rely on the detection of adjacent tetrahedra an algorithm that performs this identification and builds the set of all possible combinations of tetrahedral elements of an input mesh T into hexahedra,…
We devise an algorithm which allows one to count the number of Killing vectors for a Lorentzian manifold of dimension 3. Our algorithm relies on the principal traces of powers of the Ricci tensor and branches intricately according to the…
In this paper, we investigate the geometry of moduli space $P_d$ of degree $d$ del Pezzo pair, that is, a del Pezzo surface $X$ of degree $d$ with a curve $C \sim -2K_X$. More precisely, we study compactifications for $P_d$ from both…
The projective linear group \(\pgl(\comp,4)\) acts on cubic surfaces, considered as points of $\mathbb{P}_{\mathbb{C}}^{19}$. We compute the degree of the $15$-dimensional projective variety given by the Zariski closure of the orbit of a…