Normally generated line bundles on general curves, II

Let C be a general curve of genus g≥3. Here we prove that there is a normally generated L∈Pic^d(C) such that h⁰(C,L)=r+1≥4 (i.e. a very ample line bundle which embeds C in P^r as a projectively normal curve) if and only if (r+1)h¹≤g≤r(r−1)/2+2h¹, where h¹?g+r−d=h¹(C,L).     

On the connectedness of the scheme of multisecants to a projective curve

Here we study the connectedness of the scheme of multisecant linear spaces to a curveC P ⁿ . In particular we prove the connectedness of the scheme of trisecant lines of a smoothC P ³ with non-special hyperplane section.     

Curves in the double plane

We study in detail locally Cohen-Macaulay curves in P⁴ which are contained in a double plane 2H, thus completing the classification of curves lying on surfaces of degree two. We describe the irreducible components of the Hilbert schemes H _d,g(2H) of lo-cally Cohen-Macaulay curves in 2H of degree d and arithmetic genus g, and we show that H _d,g(2H) is connected. We also discuss the Rao module of these curves and liaison and biliaison equiva-lence classes.     

A new topological degree theory for perturbations of the sum of two maximal monotone operators

Let X be an infinite dimensional real reflexive Banach space with dual space X^∗ and G⊂X, open and bounded. Assume that X and X^∗ are locally uniformly convex. Let T:X⊃D(T)→2^X∗ be maximal monotone and strongly quasibounded, S:X⊃D(S)→X^∗ maximal monotone, and C:X⊃D(C)→X^∗ strongly quasibounded w.r.t. S and such that it satisfies a generalized (S₊)-condition w.r.t. S. Assume that D(S)=L⊂D(T)∩D(C), where L is a dense subspace of X, and 0∈T(0),S(0)=0. A new topological degree theory is introduced for the sum T+S+C, with degree mapping d(T+S+C,G,0). The reason for this development is the creation of a useful tool for the study of a class of time-dependent problems involving three operators. This degree theory is based on a degree theory that was recently developed by Kartsatos and Skrypnik just for the single-valued sum S+C, as above.     

Stratification of the Hilbert scheme of space curves with a stable normal bundle

Ford≥3g and 1≤s≤[g/2], we study the strataN _{d, g(s)} of degreed genusg spaces curvesC whose normal bundleN _C is stable with stability degree (integer of Lange-Narasimhan) σ(N _C)=2s. We prove thatN _{d, g(s)} has an irreducible component of the right dimension whose general curve has a normal bundle with the right number of maximal subbundles. We consider also the semi-stable case (s=0), obtaining similar results. We prove our results by studying the normal bundles of reducible curves and their deformations. Both authors were partially supported by MIUR and GNSAGA of INdAM (Italy).     

Remarks on the nef cone on symmetric products of curves

Let C be a very general curve of genus g and let C ⁽²⁾ be its second symmetric product. This paper concerns the problem of describing the convex cone of all numerically effective -divisors classes in the Néron–Severi space . In a recent work, Julius Ross improved the bounds on in the case of genus five. By using his techniques and by studying the gonality of the curves lying on C ⁽²⁾, we give new bounds on the nef cone of C ⁽²⁾ when C is a very general curve of genus 5 ≤ g ≤ 8. This work has been partially supported by (1) PRIN 2007 “Spazi di moduli e teorie di Lie”; (2) Indam (GNSAGA); (3) FAR 2008 (PV) “Varietá algebriche, calcolo algebrico, grafi orientati e topologici”.     

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).     

On Chow and Brauer groups of a product of Mumford curves

Let C₁,···,C_d be Mumford curves defined over a finite extension of and let X=C₁×···×C_d. We shall show the following: (1) The cycle map CH₀(X)/n → H^2d(X, μ_n^⊗d) is injective for any non-zero integer n. (2) The kernel of the canonical map CH₀(X)→Hom(Br(X),) (defined by the Brauer-Manin pairing) coincides with the maximal divisible subgroup in CH₀(X).     

The generalized Verschiebung map for curves of genus 2

Let C be a smooth curve, and M _r (C) the coarse moduli space of vector bundles of rank r and trivial determinant on C. We examine the generalized Verschiebung map induced by pulling back under Frobenius. Our main result is a computation of the degree of V ₂ for a general C of genus 2, in characteristic p > 2. We also give several general background results on the Verschiebung in an appendix.This paper was partially supported by fellowships from the National Science Foundation and Japan Society for the Promotion of Sciences.     

A characterization of bielliptic curves and applications to their moduli spaces

We deal with the covers of degree 4 naturally associated to a bielliptic curve of genus g≥6, giving a proof of the unirationality of the moduli space ? _g ^be of such curves, of the rationality of the Hurwitz scheme ℌ ^be _{4, g} of bielliptic curves of even genus g, whereas, when g is odd, we construct a finite map ℂ^{2 g -2}→? _g ^be and compute its degree. Received: March 25, 2000; in final form: March 10, 2001?Published online: May 29, 2002     

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.     

Galois points for a plane curve in characteristic two

Let C$C$ be an irreducible plane curve. A point P$P$ in the projective plane is said to be Galois with respect to C$C$ if the function field extension induced by the projection from P$P$ is Galois. We denote by δ^′(C)${\delta }^{\prime }\left(C\right)$ the number of Galois points contained in P²?C${\mathbb{P}}^2?C$. In this article we will present two results with respect to determination of δ^′(C)${\delta }^{\prime }\left(C\right)$ in characteristic two. First we determine δ^′(C)${\delta }^{\prime }\left(C\right)$ for smooth plane curves of degree a power of two. In particular, we give a new characterization of the Klein quartic in terms of δ^′(C)${\delta }^{\prime }\left(C\right)$. Second we determine δ^′(C)${\delta }^{\prime }\left(C\right)$ for a generalization of the Klein quartic, which is related to an example of Artin–Schreier curves whose automorphism group exceeds the Hurwitz bound. This curve has many Galois points.     

Fundamental groups of the complements of certain plane non-tame torus sextics

The moduli space of torus sextics with the configuration of singularities {A₂+A₅+2E₆} has two connected components. We compute the fundamental groups π₁(CP²−C) for sextics C in both components and study their differences.     

Some moduli stacks of symplectic bundles on a curve are rational

Let C be a smooth projective curve of genus g?2 over a field k. Given a line bundle L on C, let Sympl2n,L be the moduli stack of vector bundles E of rank 2n on C endowed with a nowhere degenerate symplectic form up to scalars. We prove that this stack is birational to BGm×As for some s if deg(E)=n⋅deg(L) is odd and C admits a rational point P∈C(k) as well as a line bundle ξ of degree 0 with ξ⊗2?OC. It follows that the corresponding coarse moduli scheme of Ramanathan-stable symplectic bundles is rational in this case.     

Linear series computing the Clifford index of a projective curve

For a (smooth irreducible) curveC of genus g and Clifford indexc>2 with a linear seriesg _d ^r computing c (so ) it is well known thatc + 2 ≤d ≤2 (c + 2), and if then 2c + 1 ≤g ≤ 2c + 4 unlessd = 2c + 4 in which caseg = 2c + 5. Let c ≥ 0 andg be integers. If 2c + 1 ≤g ≤2c + 4 we prove that for any integerd <g such thatd ≡c mod 2 andc + 2 ≤d < 2(c + 2) there exists a curve of genus g and Clifford index c with a g_d ^r computing c. Ford ≥c + 6 (i.e.r ≥ 3) we construct this curve on a surface of degree 2r-2 in ℙ^r, and ford ≥c + 8 (i.e.r ≥ 4) we show that such a curve cannot be found on a surface in ℙ^r of smaller degree. In fact, if g_d ^r computes the Clifford index c of C such thatc + 8 ≤d ≤ 2c + 3 then the birational morphism defined by this series cannot map C onto a (maybe, singular) curve contained in a surface of degree at most 2r-3 in ℙ^r.     

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).