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Enumerative Geometry of Plane Curve of Low Genus
"... Abstract. We collect various known results (about plane curves and the moduli space of stable maps) to derive new recursive formulas enumerating low genus plane curves of any degree with various behaviors. Recursive formulas are given for the characteristic numbers of rational plane curves, elliptic ..."
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Abstract. We collect various known results (about plane curves and the moduli space of stable maps) to derive new recursive formulas enumerating low genus plane curves of any degree with various behaviors. Recursive formulas are given for the characteristic numbers of rational plane curves, elliptic plane curves, and elliptic plane curves with fixed complex structure. Recursions are also given for the number of elliptic (and rational) plane curves with various “codimension 1 ” behavior (cuspidal, tacnodal, triple pointed, etc., as well as the geometric and arithmetic sectional genus of the Severi variety). We compute the latter numbers for genus 2 and 3 plane curves as well. We rely on results of Caporaso, Diaz, Getzler, Harris, Ran, and especially Pandharipande. 1.
On the geometry of some strata of unisingular curves
, 2008
"... We study geometric properties of linear strata of unisingular curves. We resolve the singularities of closures of the strata and represent the resolutions as projective bundles. This enables us to study their geometry. In particular we calculate the Picard groups of the strata and the intersection ..."
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We study geometric properties of linear strata of unisingular curves. We resolve the singularities of closures of the strata and represent the resolutions as projective bundles. This enables us to study their geometry. In particular we calculate the Picard groups of the strata and the intersection rings of the closures of the strata. The rational equivalence classes of some geometric cycles on the strata are calculated. As an application we give an example when the proper stratum is not affine. As an auxiliary problem we discuss the collision of two singular points, restrictions on possible resulting singularity types and solve the collision problem in several cases. Then we present some cases of enumeration of
1 On the enumeration of complex plane curves with two singular points
, 2008
"... We study equisingular strata of curves with two singular points of prescribed types. The method of the previous work [Kerner04] is generalized to this case. This allows to solve the enumerative problem for plane curves with two singular points of linear singularity types. In the general case this r ..."
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We study equisingular strata of curves with two singular points of prescribed types. The method of the previous work [Kerner04] is generalized to this case. This allows to solve the enumerative problem for plane curves with two singular points of linear singularity types. In the general case this reduces the enumerative questions to the problem of collision of the two singular points. The method is applied to several cases, e.g. enumeration of curves with two ordinary multiple points, with a point of a linear singularity type and a node etc. Explicit numerical results are given. An elementary application of the method is the determination of Thom polynomials for curves with one singular point (for some series of singularity types). Some examples are given. MSC: primary14N10, 14N35 secondary14H10, 14H50
ON A THEOREM OF HARER
, 1996
"... Let Mg be the coarse moduli space of smooth projective complex curves of genus g. Let Pic(Mg) denote the Picard group of line bundles on Mg. According to Mumford, Pic(Mg) ⊗ Q ≃ H 2 (Mg, Q) for g ≥ 2 [9]. The purpose of this note is to give another proof of the following fundamental ..."
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Let Mg be the coarse moduli space of smooth projective complex curves of genus g. Let Pic(Mg) denote the Picard group of line bundles on Mg. According to Mumford, Pic(Mg) ⊗ Q ≃ H 2 (Mg, Q) for g ≥ 2 [9]. The purpose of this note is to give another proof of the following fundamental
Picard groups of Hilbert schemes of Curves
, 1993
"... Abstract: We calculate the Picard group, over the integers, of the Hilbert scheme of smooth, irreducible, nondegenerate curves of degree d and genus g ≥ 4 in Pr, in the case when d ≥ 2g + 1 and r ≤ d − g. We express the classes of the generators in terms of some “natural ” divisor classes. Notation ..."
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Abstract: We calculate the Picard group, over the integers, of the Hilbert scheme of smooth, irreducible, nondegenerate curves of degree d and genus g ≥ 4 in Pr, in the case when d ≥ 2g + 1 and r ≤ d − g. We express the classes of the generators in terms of some “natural ” divisor classes. Notation and conventions M 0 g: Moduli space of smooth, irreducible curves of genus g ≥ 4, without automorphisms. π: Cg − → M0 g: Universal curve over M0 g. Jd (C) : Jacobian variety which parametrizes line bundles of degree d on the curve C.