General Curve Of Genus Research Articles

LINEAR SERIES ON GENERAL CURVES WITH PRESCRIBED INCIDENCE CONDITIONS

AbstractUsing degeneration and Schubert calculus, we consider the problem of computing the number of linear series of given degree d and dimension r on a general curve of genus g satisfying prescribed incidence conditions at n points. We determine these numbers completely for linear series of arbitrary dimension when d is sufficiently large, and for all d when either $r=1$ or $n=r+2$ . Our formulas generalise and give new proofs of recent results of Tevelev and of Cela, Pandharipande and Schmitt.

Cohomology of the moduli space of non-hyperelliptic genus four curves

We compute the intersection Betti numbers of the GIT model of the moduli space of Brill-Noether-Petri general curves of genus 4. This space was shown to be the final non-trivial log canonical model for the moduli space of stable genus four curves, under the Hassett-Keel program. The strategy of the cohomological computation relies on a general method developed by Kirwan to calculate the cohomology of GIT quotients of projective varieties, based on the equivariantly perfect stratification of the unstable points studied by Hesselink and others and a partial resolution of singularities.

Geometry of Prym varieties for certain bielliptic curves of genus three and five

We construct two pencils of bielliptic curves of genus three and genus five. The first pencil is associated with a general abelian surface with a polarization of type $(1,2)$. The second pencil is related to the first by an unramified double cover, the Prym variety of which is canonically isomorphic to the Jacobian of a very general curve of genus two. Our results are obtained by analyzing suitable elliptic fibrations on the associated Kummer surfaces and rational double covers among them.

Secant planes of a general curve via degenerations

We study linear series on a general curve of genus g, whose images are exceptional with respect to their secant planes. Each such exceptional secant plane is algebraically encoded by an included linear series, whose number of base points computes the incidence degree of the corresponding secant plane. With enumerative applications in mind, we construct a moduli scheme of inclusions of limit linear series with base points over families of nodal curves of compact type, which we then use to compute combinatorial formulas for the number of secant-exceptional linear series when the spaces of linear series and of inclusions are finite.

Failure of the integral Hodge conjecture for threefolds of Kodaira dimension zero

We prove that the product of an Enriques surface and a very general curve of genus at least 1 does not satisfy the integral Hodge conjecture for 1-cycles. This provides the first examples of smooth projective complex threefolds of Kodaira dimension zero for which the integral Hodge conjecture fails, and the first examples of non-algebraic torsion cohomology classes of degree 4 on smooth projective complex threefolds.

The Maximal Rank Conjecture for sections of curves

Let C⊂Pr be a general curve of genus g embedded via a general linear series of degree d. The Maximal Rank Conjecture asserts that the restriction maps H0(OPr(m))→H0(OC(m)) are of maximal rank; this determines the Hilbert function of C.In this paper, we prove an analogous statement for the union of hyperplane sections of general curves. More specifically, if H⊂Pr is a general hyperplane, and C1,C2,…,Cn are general curves, we show H0(OH(m))→H0(O(C1∪C2∪⋯∪Cn)∩H(m)) is of maximal rank, except for some counterexamples when m=2.As explained in [5], this result plays a key role in the author's proof of the Maximal Rank Conjecture [7].

On rational maps from the product of two general curves

This paper treats the dominant rational maps from the product of two very general curves to nonsingular projective surfaces. Combining the result by Bastianelli and Pirola, we prove that the product of two very general curves of genus $g\geq 7$ and $g'\geq 3$ does not admit dominant rational maps of degree $> 1$ if the image surface is non-ruled. We also treat the case of the 2-symmetric product of a curve.

Resolutions of general canonical curves on rational normal scrolls

Let $C\subset \mathbb{P}^{g-1}$ be a general curve of genus $g$ and let $k$ be a positive integer such that the Brill-Noether number $\rho(g,k,1)\geq 0$ and $g > k+1$. The aim of this short note is to study the relative canonical resolution of $C$ on a rational normal scroll swept out by a $g^1_k=|L|$ with $L\in W^1_k(C)$ general. We show that the bundle of quadrics appearing in the relative canonical resolution is unbalanced if and only if $\rho>0$ and $(k-\rho-\frac{7}{2})^2-2k+\frac{23}{4}>0$.

On dominant rational maps from products of curves to surfaces of general type

In this paper we investigate the existence of generically nite dominant rational maps from products of curves to surfaces of general type. We prove that the product C D of two distinct very general curves of genus g 7 and g 0 2 does not admit dominant rational maps|other than the identity|on surfaces of general type.

Uniruledness of some moduli spaces of stable pointed curves

We prove uniruledness of some moduli spaces M¯g,n of stable curves of genus g with n marked points using linear systems on nonsingular projective surfaces containing the general curve of genus g. Precisely we show that M¯g,n is uniruled for g=12 and n≤5, g=13 and n≤3, g=15 and n≤2.We then prove that the pointed hyperelliptic locus Hg,n is uniruled for g≥2 and n≤4g+4.In the last part we show that a nonsingular complete intersection surface does not carry a linear system containing the general curve of genus g≥16 and if it carries a linear system containing the general curve of genus 12≤g≤15, then it is canonical.

A note on Griffiths infinitesimal invariant for curves

Given a generic curve of genus $g\geqslant4$ and a smooth point $L\in W_{g-1}^{1}(C)$, whose linear system is base-point free, we consider the Abel-Jacobi normal function associated to $L^{\otimes 2}\otimes \omega_{C}^{-1}$, when $(C,L)$ varies in moduli. We prove that its infinitesimal invariant reconstruct the couple $(C,L)$. When $g=4$, we obtain the generic Torelli theorem proved by Griffiths.

On symmetric products of curves

Let $C$ be a smooth complex projective curve of genus $g$ and let $C^{(2)}$ be its second symmetric product. This paper concerns the study of some attempts at extending to $C^{(2)}$ the notion of gonality. In particular, we prove that the degree of irrationality of $C^{(2)}$ is at least $g-1$ when $C$ is generic and that the minimum gonality of curves through the generic point of $C^{(2)}$ equals the gonality of $C$. In order to produce the main results we deal with correspondences on the $k$-fold symmetric product of $C$, with some interesting linear subspaces of $\mathbb {P}^n$ enjoying a condition of Cayley-Bacharach type, and with monodromy of rational maps. As an application, we also give new bounds on the ample cone of $C^{(2)}$ when $C$ is a generic curve of genus ${6\leq g\leq 8}$.

ON AN EXAMPLE OF MUKAI

AbstractIn this paper we use an example of Mukai to construct semistable bundles of rank 3 with six independent sections on a general curve of genus 9 or 11 with Clifford index strictly less than the Clifford index of the curve. The example also allows us to show the non-emptiness of some Brill–Noether loci with negative expected dimension.

Kernel Bundles, Syzygies of Points, and the Effective Cone of Cg-2

We obtain a complete description of the effective cone of C g−2 when C is a general curve of genus g ≥ 6, as well as a new bound in the case where C is a smooth plane quintic. In addition, we obtain a new virtual bound for the effective cone of C g−2m which is a genuine bound when m = 2, and we also characterize certain natural divisors on C g−2 as subordinate loci associated to adjunctions of kernel bundles.

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 h 0 ( 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 ) h 1 ≤ g ≤ r ( r − 1 ) / 2 + 2 h 1 , where h 1 ≔ g + r − d = h 1 ( C , L ) .

Geometry of curves with exceptional secant planes: linear series along the general curve

We study linear series on a general curve of genus g, whose images are exceptional with regard to their secant planes. Working in the framework of an extension of Brill–Noether theory to pairs of linear series, we prove that a general curve has no linear series with exceptional secant planes, in a very precise sense, whenever the total number of series is finite. Next, we partially solve the problem of computing the number of linear series with exceptional secant planes in a one-parameter family in terms of tautological classes associated with the family, by evaluating our hypothetical formula along judiciously-chosen test families. As an application, we compute the number of linear series with exceptional secant planes on a general curve equipped with a one-dimensional family of linear series. We pay special attention to the extremal case of d-secant (d − 2)-planes to (2d − 1)-dimensional series, which appears in the study of Hilbert schemes of points on surfaces. In that case, our formula may be rewritten in terms of hypergeometric series, which allows us both to prove that it is nonzero and to deduce its asymptotics in d.

Vectors bundles with theta divisors I—Bundles on Castelnuovo curves

In this paper we show that semistable vector bundles on a Castelnuovo curve of genus g ≥ 2 have theta divisors. As a corollary, we deduce that semistable vector bundles on a smooth, general curve of genus g ≥ 2 which extend to semistable vector bundles on any flat Castelnuovo degeneration of the general curve admit a theta divisor.

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 $${Nef\,(C^{(2)})_{\mathbb{R}}}$$ of all numerically effective $${\mathbb{R}}$$ -divisors classes in the Néron–Severi space $${N^1(C^{(2)})_{\mathbb{R}}}$$ . In a recent work, Julius Ross improved the bounds on $${Nef\,(C^{(2)})_{\mathbb{R}}}$$ 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.

On nondegeneracy of curves

We study the conditions under which an algebraic curve can be modelled by a Laurent polynomial that is nondegenerate with respect to its Newton polytope. We prove that every curve of genus g ≤ 4 over an algebraically closed field is nondegenerate in the above sense. More generally, let M g be the locus of nondegenerate curves inside the moduli space of curves of genus g ≥ 2. Then we show that dimM g = min(2g +1, 3g − 3), except for g = 7 where dimM 7 = 16; thus, a generic curve of genus g is nondegenerate if and only if g ≤ 4. Subject classification: 14M25, 14H10 Let k be a perfect field with algebraic closure k. Let f ∈ k[x, y] be an irreducible Laurent polynomial, and write f = ∑ (i,j)∈Z2 cijx y . We denote by supp(f) = {(i, j) ∈ Z : cij 6= 0} the support of f , and we associate to f its Newton polytope ∆ = ∆(f), the convex hull of supp(f) in R. We assume throughout that ∆ is 2-dimensional. For a face τ ⊂ ∆, let f |τ = ∑ (i,j)∈τ cijx y . We say that f is nondegenerate if, for every face τ ⊂ ∆ (of any dimension), the system of equations (1) f |τ = x ∂f |τ ∂x = y ∂f |τ ∂y = 0 has no solutions in k ∗2 . From the perspective of toric varieties, the condition of nondegeneracy can be rephrased as follows. The Laurent polynomial f defines a curve U(f) in the torus Tk = Spec k[x , y], and Tk embeds canonically in the projective toric surface X(∆)k associated to ∆ over k. Let V (f) be the Zariski closure of the curve U(f) inside X(∆)k. Then f is nondegenerate if and only if for every face τ ⊂ ∆, we have that V (f)∩Tτ is smooth of codimension 1 in Tτ , where Tτ is the toric component of X(∆)k associated to τ . (See Proposition 1.2 for alternative characterizations.) Nondegenerate polynomials have become popular objects in explicit algebraic geometry, owing to their connection with toric geometry [4]: a wealth of geometric information about V (f) is contained in the combinatorics of the Newton polytope ∆(f). The notion was initially employed by Kouchnirenko [22], who studied nondegenerate polynomials in the context of singularity theory. Nondenegerate polynomials emerge naturally in the theory of sparse resultants [14] and admit a linear effective Nullstellensatz [8, Section 2.3]. They make an appearance in the study of real algebraic curves in maximal position [26] and in the problem of enumerating curves through a set of prescribed points [27]. In the case where k is a finite field, they arise in the construction of curves with many points [6, 23], in the p-adic cohomology theory of Adolphson and Sperber [2], and in explicit methods for computing zeta functions of varieties over k [8]. Despite their utility and seeming Date: 26 December 2008.

Obstructions to deforming curves on a 3-fold, I: A generalization of Mumford’s example and an application to 𝐻𝑜𝑚 schemes

We give a sufficient condition for a first order infinitesimal deformation of a curve on a 3 3 -fold to be obstructed. As application we construct generically non-reduced components of the Hilbert schemes of uniruled 3 3 -folds and the Hom scheme from a general curve of genus five to a general cubic 3 3 -fold.