On a class of rational cuspidal plane curves

We obtain new examples and the complete list of the rational cuspidal plane curvesC with at least three cusps, one of which has multiplicitydegC-2. It occurs that these curves are projectively rigid. We also discuss the general problem of projective rigidity of rational cuspidal plane curves.     

Counting generic genus- curves on Hirzebruch surfaces

Hirzebruch surfaces provide an excellent example to underline the fact that in general symplectic manifolds, Gromov-Witten invariants might well count curves in the boundary components of the moduli spaces. We use this example to explain in detail that the counting argument given by Batyrev for toric manifolds does not work.

     

The effect of points fattening on Hirzebruch surfaces

The purpose of this note is to study initial sequences of 0–dimensional subschemes of Hirzebruch surfaces and classify subschemes whose initial sequence has the minimal possible growth.     

On the number of moduli of plane sextics with six cusps

Let be the variety of irreducible sextics with six cusps as singularities. Let be one of irreducible components of . Denoting by the space of moduli of smooth curves of genus 4, we consider the rational map sending the general point [Γ] of Σ, corresponding to a plane curve , to the point of parametrizing the normalization curve of Γ. The number of moduli of Σ is, by definition the dimension of Π(Σ). We know that , where ρ(2, 4, 6) is the Brill–Noether number of linear series of dimension 2 and degree 6 on a curve of genus 4. We prove that both irreducible components of have number of moduli equal to seven.      

Adjoint divisors on algebraic curves

Let C be a numerically connected curve lying on a smooth algebraic surface. We show that if is an ample invertible sheaf satisfying some technical numerical hypotheses then is normally generated. As a corollary we show that the sheaf ω_C^⊗2 on a numerically connected curve C of arithmetic genus p_a?3 is normally generated if ω_C is ample and does not exist a subcurve B⊂C such that p_a(B)=1=B(C−B).     

The minimal degree of plane models of double covers of smooth curves

If X is a smooth curve such that the minimal degree of its plane models is not too small compared with its genus, then X has been known to be a double cover of another smooth curve Y under some mild condition on the genera. However there are no results yet for the minimal degrees of plane models of double covers except some special cases. In this paper, we give upper and lower bounds for the minimal degree of plane models of the double cover X in terms of the gonality of the base curve Y and the genera of X and Y. In particular, the upper bound equals to the lower bound in case Y is hyperelliptic. We give an example of a double cover which has plane models of degree equal to the lower bound.     

Singular rational curves with points of nearly-maximal weight

In this article we study rational curves with a unique unibranch genus-g singularity, which is of κ-hyperelliptic type in the sense of [27]; we focus on the cases $\kappa =0$ and $\kappa =1$, in which the semigroup associated to the singularity is of (sub)maximal weight. We obtain a partial classification of these curves according to the linear series they support, the scrolls on which they lie, and their gonality.     

The universal theta divisor over the moduli space of curves

By computing the class of the universal antiramification locus of the Gauss map, we obtain a complete birational classification by Kodaira dimension of the universal theta divisor over the moduli space of curves.     

On projectively normal embeddings of generalk-gonal curves

Fix integersg, k andt witht>0,k≥3 andtk<g/2−1. LetX be a generalk-gonal curve of genusg andR∈Pic ^k (X) the uniqueg _k ¹ onX. SetL:=K _X⊗(R ^*)^⊗t.L is very ample. Leth _L:X→P(H ⁰(X, L)^*) be the associated embedding. Here we prove thath _L(X) is projectively normal. Ifk≥4 andtk<g/2−2 the curveh _L(X) is scheme-theoretically cut out by quadrics. The author was partially supported by MURST and GNSAGA of CNR (Italy).     

Rational cuspidal plane curves of type (d,d-4) with

In the present paper we classify rational cuspidal plane curves with maximal multiplicity deg C - 4 and at least three cusps and where (V,D) is the minimal (SNC) resolution of (ℙ²,C). Received: 28 August 1998     

On fundamental groups related to the Hirzebruch surface F_1

Given a projective surface and a generic projection to the plane,the braid monodromy factorization(and thus,the braid monodromy type)of the complement of its branch curve is one of the most important topological invariants,stable on deformations.From this factorization,one can compute the fundamental group of the complement of the branch curve,either in C~2 or in CP~2.In this article,we show that these groups,for the Hirzebruch surface F_1,(a,b),are almost-solvable.That is, they are an extension of a solvable group,which strengthen the conjecture on degeneratable surfaces.     

On certain remarkable curves of genus five

The aim of this note is twofold. First to show the existence of genus five curves having exactly twenty four Weierstrass points, which constitute the set of fixed points of three distinct elliptic involutions on them. Second to characterize these curves, in fact we prove that all such curves are bielliptic double cover of Fermat's quartic.     

On unramified normal coverings of hyperelliptic curves

It is well known that the number of unramified normal coverings of an irreducible complex algebraic curve C with a group of covering transformations isomorphic to Z₂⊕Z₂ is (2^4g−3⋅2^2g+2)/6. Assume that C is hyperelliptic, say . Horiouchi has given the explicit algebraic equations of the subset of those covers which turn out to be hyperelliptic themselves. There are of this particular type. In this article, we provide algebraic equations for the remaining ones.     

Self-adjoint determinantal representations of real plane curves

Research supported by a Chaim Weizmann Fellowship, Weizmann Institute of Science, Rehovot, Israel.     

On the gonality of stable curves

In this paper we use admissible covers to investigate the gonality of a stable curve C over $\mathbb{C}$. If C is irreducible, we compare its gonality to that of its normalization. If C is reducible, we compare its gonality to that of its irreducible components. In both cases we obtain lower and upper bounds. Furthermore, we show that four admissible covers constructed give rise to generically injective maps between Hurwitz schemes. We show that the closures of the images of three of these maps are components of the boundary of the target Hurwitz schemes, and the closure of the image of the remaining map is a component of a certain codimension‐1 subscheme of the boundary of the target Hurwitz scheme.     

Topology of the compactified Jacobians of singular curves

We compute the Euler number of the compactified Jacobian of a curve whose minimal unibranched normalization has only plane irreducible singularities with characteristic Puiseux exponents (p, q), (4, 2q, s), (6, 8, s), or (6, 10, s). Further, we derive a combinatorial method to compute the Betti numbers of the compactified Jacobian of an unibranched rational curve with singularities like above. Some of the Betti numbers can be stated explicitly.