Divisors on symmetric products of curves

Kouvidakis, Alexis

doi:10.1090/s0002-9947-1993-1149124-5

Cited by 29 publications

(27 citation statements)

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“…the nef cone is as large as possible. Such a conjecture has been proved in [4,13] when the genus g is a perfect square, whereas the problem is still open in the other cases. We recall further that when the genus of C is g ≥ 9, Kouvidakis' conjecture is implied by Nagata's on the Seshadri constant at g generic points in P 2 , and this fact leads to several bounds on τ (C) (see for instance [19]).…”

Section: Bounds On the Ample Cone Of Second Symmetric Products Of Curvesmentioning

confidence: 99%

On symmetric products of curves

Bastianelli

2012

Trans. Amer. Math. Soc.

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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 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 ≤ g ≤ 8

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Section: Bounds On the Ample Cone Of Second Symmetric Products Of Curvesmentioning

confidence: 99%

On symmetric products of curves

Bastianelli

2012

Trans. Amer. Math. Soc.

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“…H ≡ D ′ a,k , to be very ample. In this section we use a description of the nef cones on symmetric squares given in Kouvidakis [21] together with Reider's characterization of very ampleness [38] to show that H is very ample in the following cases:…”

Section: Divisors On Cartesian and Symmetric Squares Of Curves Of Genmentioning

confidence: 99%

TWO CLASSES OF HYPERBOLIC SURFACES IN ℙ³

Shiffman

Zaidenberg

2000

Int. J. Math.

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We construct two classes of singular Kobayashi hyperbolic surfaces in P 3 . The first consists of generic projections of the cartesian square V = C × C of a generic genus g ≥ 2 curve C smoothly embedded in P 5 . These surfaces have C-hyperbolic normalizations; we give some lower bounds for their degrees and provide an example of degree 32. The second class of examples of hyperbolic surfaces in P 3 is provided by generic projections of the symmetric square V ′ = C 2 of a generic genus g ≥ 3 curve C. The minimal degree of these surfaces is 16, but this time the normalizations are not C-hyperbolic.Proposition 1.1. Let X be a reduced compact complex space, and let π : X → X be the normalization of X. Assume that the space X is Kobayashi hyperbolic and let S ⊂ X, resp. S := π(S) ⊂ X, be the ramification divisor, resp. the branching divisor, so that the restriction π | (X \ S) : X \ S → X \ S is biholomorphic. Then X is Kobayashi hyperbolic if and only if S is Kobayashi hyperbolic.Proof. Clearly, if the space X is hyperbolic, then so is the subspace S of X. By the Brody Theorem [5] (see [18], [20, (3.6.3)] or [45] for the case of complex spaces), the compact complex space X is hyperbolic iff any holomorphic mapping f : C → X is constant. Assuming that the branching divisor S is hyperbolic, we may restrict the consideration to the mappings f : C → X with the image not contained in S. In this case f can be lifted to X (see [34]) and hence, in virtue of the hyperbolicity of the complex space X, it is constant.Applying Proposition 1.1 to the case where V ⊂ P N , N ≥ 4, is a smooth Kobayashi hyperbolic surface (e.g., V can be isomorphic to the cartesian product of two smooth projective curves Γ 1 and Γ 2 of genera g 1 , g 2 ≥ 2) and V ⊂ P 3 is a generic projection of V with the irreducible double curve S, we obtain the following statement: Corollary 1.2. Let V ⊂ P 3 be a generic projection of a Kobayashi hyperbolic smooth projective surface. Then V is Kobayashi hyperbolic iff the double curve S is Kobayashi hyperbolic; i.e., iff the geometric genus of S is at least 2.

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“…There has been a great deal of interest in understanding the various positive cones of curves and divisors on algebraic varieties. Several cases have been analyzed, including symmetric products of curves in [6], [8], abelian varieties in [1], [3], and holomorphic symplectic varieties in [5], [2]. The main result of this paper is the following theorem.…”

Section: Introductionmentioning

confidence: 99%

Some surfaces with non-polyhedral nef cones

Rabindranath

2018

Proc. Amer. Math. Soc.

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Divisors on symmetric products of curves

Abstract: Abstract.For a curve with general moduli, the Neron-Severi group of its symmetric products is generated by the classes of two divisors x and 6 . In this paper we give bounds for the cones of eifective and ample divisors in the x8-plane.

Cited by 29 publications

References 1 publication

On symmetric products of curves

On symmetric products of curves

TWO CLASSES OF HYPERBOLIC SURFACES IN ℙ³

Some surfaces with non-polyhedral nef cones

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Divisors on symmetric products of curves

Abstract: Abstract.For a curve with general moduli, the Neron-Severi group of its symmetric products is generated by the classes of two divisors x and 6 . In this paper we give bounds for the cones of eifective and ample divisors in the x8-plane.

Cited by 29 publications

References 1 publication

On symmetric products of curves

On symmetric products of curves

TWO CLASSES OF HYPERBOLIC SURFACES IN ℙ3

Some surfaces with non-polyhedral nef cones

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TWO CLASSES OF HYPERBOLIC SURFACES IN ℙ³