Related papers: Hyperelliptic plane curves of type (d,d-2)

The `linear orbit' of a plane curve of degree d is its orbit in the projective space of dimension d(d+3)/2 parametrizing such curves under the natural action of PGL(3). In this paper we compute the degree of the closure of the linear orbits…

We construct genus one curves on base extensions of generic Severi--Brauer varieties of a given index and period which are versal objects for families of geometrically elliptic normal curves. We also compute the periods and indices of these…

In this article, we are interested in finding rational points on certain superelliptic curves.

We investigate the number and the geometry of smooth hyperelliptic curves on a general complex abelian surface. We show that the only possibilities of genera of such curves are $2,3,4$ and $5$. We focus on the genus 5 case. We prove that up…

Let S be a smooth, projective surface of Picard rank 1 and very ample generator embedding S into P^n. Let C be a smooth curve in O(m) for m \geq 5. We prove that any base-point free, complete g^r_d on C for r\in\{1,2\} and d small enough is…

In this paper we describe projective curves and surfaces such that almost all their hyperplane sections are projectively equivalent. Our description is complete for curves and close to being complete for smooth surfaces. In the appendix we…

In this paper we consider the question of whether there exists a hyperelliptic curve of genus $g$ which is defined over $\FF_q$ but has no rational points over $\FF_q$ for various pairs $(g,q)$.

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 characterize the equisingularity classes of irreducible plane curve germs whose general members have a Newton nondegenerate general polar curve. In addition, we give explicit Zariski open sets of curves in such equisingularity classes…

We generalize the group law of curves of degree three by chords and tangents to the Jacobi variety of a hyperelliptic curve. In the case of genus 2 we accomplish the construction by a cubic parabola. We derive explicit rational formulas for…

For a given hyperelliptic curve $C$ over a finite field with Jacobian $J_C$, we consider the hyperelliptic analogue of the congruential generator defined by $W_n=W_{n-1}+D$ for $n\geq 1$ and $D,W_0\in J_C$. We show that curves of genus 2…

We develop a new approach for constructing normalized differentials on hyperelliptic curves of infinite genus and obtain uniform asymptotic estimates for the distribution of their zeros.

We give a complete classification of complex Q-homology projective planes with isolated rational double point singularities and numerically trivial canonical bundle. There are 31 types, and each has one-dimensional moduli. In fact, all…

We prove that the moduli space of contact stable maps to $\mathbb{P}^{2n+1}$ of degree $d$ admits a stratification parameterized by graphs. We use it to determine the number of irreducible rational contact curves in $\mathbb{P}^{2n+1}$ with…

Cais, Ellenberg and Zureick-Brown recently observed that over finite fields of characteristic two, all sufficiently general smooth plane projective curves of a given odd degree admit a non-trivial rational 2-torsion point on their Jacobian.…

We generalize results by Wakabayashi and Orevkov about rational cuspidal curves on the projective plane to that on $\mathbb{Q}$-homology projective planes. It turns out that the result is exactly the same as the projective plane case under…

We present algorithms to compute the topology of 2D and 3D hyperelliptic curves. The algorithms are based on the fact that 2D and 3D hyperelliptic curves can be seen as the image of a planar curve (the Weierstrass form of the curve), whose…

We prove that, if two germs of plane curves $(C,0)$ and $(C',0)$ with at least one singular branch are equivalent by a (real) smooth diffeomorphism, then $C$ is complex isomorphic to $C'$ or to $\overline{C'}$. A similar result was shown by…

We give a complete classification, up to birational equivalence, of all fibrations by plane projective rational quartic curves in characteristic two.

Let $\V_{d,n}$ be the Severi variety of irreducible plane curves of degree $d\ge 4$ having $n$ nodes, with $0\le n \le \binom{d-1}{2}-1$. We prove that for every $[\ol C]\in \V_{d,n}$, the infinitesimal variation of the Hodge structure of…