Related papers: Gonality of a general ACM curve in projective 3-sp…

In this paper we characterize the irreducible curves lying in $C^{(2)}$. We prove that a curve $B$ has a degree one morphism to $C^{(2)}$ with image a curve of degree $d$ with irreducible preimage in $C\times C$ if and only if there exists…

We show that smooth curves in the same biliaison class on a hypersurface in $\mathbf{P}^3$ with ordinary singularities are linearly equivalent. We compute the invariants $h^0(\mathscr{I}_C(d))$, $h^1(\mathscr{I}_C(d))$ and…

For a smooth plane cubic $B$, we count curves $C$ of degree $d$ such that the normalizations of $C\backslash B$ are isomorphic to $\Bbb A^1$, for $d\leq7$ (for $d=7$ under some assumption). We also count plane rational quartic curves…

In this work, we describe the Cohen-Macaulay space CM of twisted cubics parameterizing curves $C$ together with a finite map $i: C \to \mathbb{P}^3$ that is generically a closed immersion and such that $C$ has Hilbert polynomial $p(t)=3t+1$…

Let $K$ be an infinite field and let $m_1,\ldots,m_n$ be a generalized arithmetic sequence of positive integers, i.e., there exist $h, d, m_1 \in\mathbb{Z}^+$ such that $m_i = h m_1 + (i-1)d$ for all $i \in \{2,\ldots,n\}$. We consider the…

Let S be a minimal complex surface of general type with irregularity q>=2 and let C be an irreducible curve of geometric genus g contained in S. Assume that C is "Albanese defective", i.e., that the image of C via the Albanese map does not…

The main purpose in this paper is to study the gonality, the Clifford index and the Clifford dimension on linearly equivalent smooth curves on Enriques surfaces. The method is similar to techniques of M.Green $\&$ R.Lazarsfeld and…

Given n general points p_1, p_2,..., p_n in P^r, it is natural to ask when there exists a curve C \subset P^r, of degree d and genus g, passing through p_1, p_2,..., p_n. In this paper, we give a complete answer to this question for curves…

We study the deformations of a smooth curve $C$ on a smooth projective threefold $V$, assuming the presence of a smooth surface $S$ satisfying $C \subset S \subset V$. Generalizing a result of Mukai and Nasu, we give a new sufficient…

Let $f\colon S\to B$ be a non-isotrivial fibred surface. We prove that the genus $g$, the rank $u_f$ of the unitary summand of the Hodge bundle $f_*\omega_f$ and the Clifford index $c_f$ satisfy the inequality $u_f \leq g - c_f$. Moreover,…

We study the Hilbert scheme of smooth, irreducible, non-degenerate and linearly normal curves of degree $d$ and genus $g$ in $\mathbb{P}^r$ ($r\ge 3$) whose complete and very ample hyperplane linear series $\mathcal{D}$ have relatively…

For $q\leq 3$ smooth plane algebraic curves $\mathcal{C}_i$ having simple normal crossings, if the invariant logarithmic $2$-jet differential bundle associated to $(\mathbb{P}^2(\mathbb{C}), \sum_{i=1}^q \mathcal{C}_i)$ has a nonzero…

In this paper we obtain bounds on $h^0(E)$ where $E$ is a semistable bundle of rank 3 over a smooth irreducible projective curve $X$ of genus $g \geq 2$ defined over an algebraically closed field of characteristic 0. These bounds are…

A curve $\gamma$ in a Riemannian manifold $M$ is three-dimensional if its torsion (signed second curvature function) is well-defined and all higher-order curvatures vanish identically. In particular, when $\gamma$ lies on an oriented…

We consider the local-global principle for divisibility in the Mordell-Weil group of a CM elliptic curve defined over a number field. For each prime $p$ we give sharp lower bounds on the degree $d$ of a number field over which there exists…

We use Green's canonical syzygy conjecture for generic curves to prove that the Green-Lazarsfeld gonality conjecture holds for generic curves of genus g, and gonality d, if $g/3<d<[g/2]+2$.

We establish for smooth projective real curves the equivalent of the classical Clifford inequality known for complex curves. We also study the cases when equality holds.

Let $C$ be a smooth curve of genus $g\ge 4$ and Clifford index $c$. In this paper, we prove that if $C$ is neither hyperelliptic nor bielliptic with $g\ge 2c+5$ and $\mathcal M$ computes the Clifford index of $C$, then either $\deg \mathcal…

A general hypersurface $X$ of degree $\leq n$ in projective space contains curves $C$ of any genus $g\geq 0$ and sufficiently large degree depnding on $g$ whose normal and conormal bundles have good postulation or natural cohomology in the…

In this paper we obtain an explicit formula for the number of curves in two dimensional complex projective space, of degree d, passing through d(d+3)/2-(k+1) generic points and having one node and one codimension k singularity, where k is…