Linear pencils on real algebraic curves

Our knowledge of linear series on real algebraic curves is still very incomplete. In this paper we restrict to pencils (complete linear series of dimension one). Let X denote a real curve of genus g with real points and let k(R) be the smallest degree of a pencil on X (the real gonality of X). Then we can find on X a base point free pencil of degree g+1 (resp. g if X is not hyperelliptic, i.e. if k(R)>2) with an assigned geometric behaviour w.r.t. the real components of X, and if we prove that which is the same bound as for the gonality of a complex curve of even genus g. Furthermore, if the complexification of X is a k-gonal curve (k≥2) one knows that k≤k(R)≤2k−2, and we show that for any two integers k≥2 and 0≤n≤k−2 there is a real curve with real points and k-gonal complexification such that its real gonality is k+n.     

Higher syzygies of hyperelliptic curves

Let X be a hyperelliptic curve of arithmetic genus g and let f:X→P¹ be the hyperelliptic involution map of X. In this paper we study higher syzygies of linearly normal embeddings of X of degree d≤2g. Note that the minimal free resolution of X of degree ≥2g+1 is already completely known. Let A=f^∗O_P¹(1), and let L be a very ample line bundle on X of degree d≤2g. For , we call the pair (m,d−2m)the factorization type ofL. Our main result is that the Hartshorne-Rao module and the graded Betti numbers of the linearly normal curve embedded by |L| are precisely determined by the factorization type of L.     

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).     

Abel maps for curves of compact type

Recently, the first Abel map for a stable curve of genus g≥2 has been constructed. Fix an integer d≥1 and let C be a stable curve of compact type of genus g≥2. We construct two d-th Abel maps for C, having different targets, and we compare the fibers of the two maps. As an application, we get a characterization of hyperelliptic stable curves of compact type with two components via the second Abel map.     

Deformations of the generalised Picard bundle

Let X be a nonsingular algebraic curve of genus g?3, and let M_ξ denote the moduli space of stable vector bundles of rank n?2 and degree d with fixed determinant ξ over X such that n and d are coprime. We assume that if g=3 then n?4 and if g=4 then n?3, and suppose further that n₀, d₀ are integers such that n₀?1 and nd₀+n₀d>nn₀(2g-2). Let E be a semistable vector bundle over X of rank n₀ and degree d₀. The generalised Picard bundle W_ξ(E) is by definition the vector bundle over M_ξ defined by the direct image where U_ξ is a universal vector bundle over X×M_ξ. We obtain an inversion formula allowing us to recover E from W_ξ(E) and show that the space of infinitesimal deformations of W_ξ(E) is isomorphic to H¹(X,End(E)). This construction gives a locally complete family of vector bundles over M_ξ parametrised by the moduli space M(n₀,d₀) of stable bundles of rank n₀ and degree d₀ over X. If (n₀,d₀)=1 and W_ξ(E) is stable for all E∈M(n₀,d₀), the construction determines an isomorphism from M(n₀,d₀) to a connected component M⁰ of a moduli space of stable sheaves over M_ξ. This applies in particular when n₀=1, in which case M⁰ is isomorphic to the Jacobian J of X as a polarised variety. The paper as a whole is a generalisation of results of Kempf and Mukai on Picard bundles over J, and is also related to a paper of Tyurin on the geometry of moduli of vector bundles.     

Normally generated line bundles on general curves

Let L be a very ample line bundle of degree d on a general curve X of genus g≥2. Here we prove that if then L is globally generated, i.e. L embeds X as a projectively normal curve in PH⁰(L).     

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.     

Some notes on graphs whose index is close to 2

We consider two classes of graphs: (i) trees of order n and diameter d =n − 3 and (ii) unicyclic graphs of order n and girth g = n − 2. Assuming that each graph within these classes has two vertices of degree 3 at distance k, we order by the index (i.e. spectral radius) the graphs from (i) for any fixed k (1 ? k ? d − 2), and the graphs from (ii) independently of k.     

Counting hyperelliptic curves

We find a closed formula for the number hyp(g) of hyperelliptic curves of genus g over a finite field k=Fq of odd characteristic. These numbers hyp(g) are expressed as a polynomial in q with integer coefficients that depend on g and the set of divisors of q−1 and q+1. As a by-product we obtain a closed formula for the number of self-dual curves of genus g. A hyperelliptic curve is defined to be self-dual if it is k-isomorphic to its own hyperelliptic twist.     

Castelnuovo-Mumford regularity for nonhyperelliptic curves

The purpose here is to show that an irreducible, reduced, projective, nonhyperelliptic curve of degree d and genus g is n-regular for if Received: 10 July 2003     

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).     

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.     

On the connectivity and superconnected graphs with small diameter

In this paper, first we prove that any graph G is 2-connected if diam(G)≤g−1 for even girth g, and for odd girth g and maximum degree Δ≤2δ−1 where δ is the minimum degree. Moreover, we prove that any graph G of diameter diam(G)≤g−2 satisfies that (i) G is 5-connected for even girth g and Δ≤2δ−5, and (ii) G is super-κ for odd girth g and Δ≤3δ/2−1.     

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.     

On the variety parameterizing completely decomposable polynomials

The purpose of this paper is to relate the variety parameterizing completely decomposable homogeneous polynomials of degree d in n+1 variables on an algebraically closed field, called , with the Grassmannian of (n−1)-dimensional projective subspaces of P^n+d−1. We compute the dimension of some secant varieties to . Moreover by using an invariant embedding of the Veronese variety into the Plücker space, we are able to compute the intersection of G(n−1,n+d−1) with , some of its secant varieties, the tangential variety and the second osculating space to the Veronese variety.     

Trigonal Gorenstein curves

We notice that the Maroni invariant of a trigonal Gorenstein curve of arithmetic genus g larger than four may be equal to zero, and we show that this happens if and only if the g₃¹ admits a non-removable base point, which is necessarily a singularity of the curve. We realize and study trigonal curves on rational scrolls, which in the case, where the g₃¹ admits a base point Q, degenerate to a cone with vertex Q.     

A bound on the Castelnuovo-Mumford regularity for curves

Recall that a projective curve in with ideal sheaf is said to be n-regular if for every integer and that in this case, it is cut out scheme-theoretically by equations of degree at most n. The purpose here is to show that an irreducible, reduced, projective curve of degree d and large arithmetic genus satisfies a smaller regularity bound than the optimal one . For example, if then a curve is -regular unless it is embedded by a complete linear system of degree . Received: 29 May 2000 / Published online: 24 September 2001