On the poles of regular differentials of singular curves

Stöhr, Karl-Otto

doi:10.1007/bf01231698

Cited by 31 publications

(20 citation statements)

References 25 publications

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“…In [29], the second author introduced a zeta function Z(Ca(X), T ) associated to the effective Cartier divisors on X. Other types of zeta functions associated to singular curves over finite fields were introduced in [15], [16], [23], [24], and [30]. The zeta function Z(Ca(X), T ) admits an Euler product with nontrivial factors at the singular points of X.…”

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confidence: 99%

Motivic zeta functions for curve singularities

Moyano‐Fernández¹,

Zúñiga–Galindo²

2010

Nagoya Mathematical Journal

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Abstract. Let X be a complete, geometrically irreducible, singular, algebraic curve defined over a field of characteristic p big enough. Given a local ring OP,X at a rational singular point P of X, we attached a universal zeta function which is a rational function and admits a functional equation if OP,X is Gorenstein. This universal zeta function specializes to other known zeta functions and Poincaré series attached to singular points of algebraic curves. In particular, for the local ring attached to a complex analytic function in two variables, our universal zeta function specializes to the generalized Poincaré series introduced by Campillo, Delgado, and Gusein-Zade. §1. Introduction Let X be a complete, geometrically irreducible, singular, algebraic curve defined over a finite field F q . In [29], the second author introduced a zeta function Z(Ca(X), T ) associated to the effective Cartier divisors on X. Other types of zeta functions associated to singular curves over finite fields were introduced in [15], [16], [23], [24], and [30]. The zeta function Z(Ca(X), T ) admits an Euler product with nontrivial factors at the singular points of X. If P is a rational singular point of X, then the local factor Z Ca(X) (T, q, O P,X ) at P is a rational function of T depending on q and the completion O P,X of the local ring O P,X of X at P . If the residue field of O P,X is not too small, then Z Ca(X) (T, q, O P,X ) depends only on the semigroup of O P,X (see [29, Theorem 5.5]). Thus, if O P,X ∼ = F q Jx, yK/(f (x, y)),

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confidence: 99%

Motivic zeta functions for curve singularities

Moyano‐Fernández¹,

Zúñiga–Galindo²

2010

Nagoya Mathematical Journal

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“…which has been shown in [8], Proposition 2.2 under the assumption that the constant field is infinite. Until now, in this section we did not make any assumption on the constant field.…”

Section: Corollary 42 the Sum And The Product Of Dim(b·mentioning

confidence: 88%

Multi-variable Poincaré series of algebraic curve singularities over finite fields

Stöhr¹

2008

Math. Z.

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Let O be the local ring at a singular point of a geometrically integral algebraic curve defined over a finite field, and let m be the number of branches centered at the curve singularity. By encoding cardinalities of certain finite sets of ideals, we associate to each pair of ideal classes of O a power series in m variables with integer coefficients, which can be represented by an integral within the framework of harmonic analysis. We prove that partial local zeta functions can be expressed in terms of these multi-variable power series. The main objective of this paper is to investigate the properties of these series, and to provide in this way a deeper insight into the nature of local zeta functions.

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“…The Examples 4 and 5 are not linearly normal. In [3], p. 244, there are two descriptions of the canonical defect D, the one due to Grothendieck (see [1]) and the one due to Rosenlicht ([11], IV.9); for Rosenlicht 's description, see [12], §2). The following examples may be checked using [12], §2.…”

Section: Proof Of Propositionmentioning

confidence: 99%

Lowest Degree Spanned Line Bundles on Integral Projective Curves

Ballico¹

2001

Demonstratio Mathematica

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IntroductionLet Y be an integral projective curve defined over an algebraically closed field K and TT : X -• Y its normalization. Set g := p a {Y) and q := p a (X). We always assume g > 4. The dualizing sheaf uiy is locally free if and only if Y is Gorenstein. We have hP{Y,uy) = g. Since g > 3, uxy is spanned by its global sections ([10]). Set L := TT*(coy)/Tors(n*(wy)).Since X is smooth, L is a line bundle. We always assume that Y is not hyperelliptic. Hence u is a birational map ([10], Th. 17). Hence C is an integral non-degenerate curve in P 9_1 and deg(C) = d. Thus C is a partial normalization of Y and q < 7 < g.

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On the poles of regular differentials of singular curves

Cited by 31 publications

References 25 publications

Motivic zeta functions for curve singularities

Motivic zeta functions for curve singularities

Multi-variable Poincaré series of algebraic curve singularities over finite fields

Lowest Degree Spanned Line Bundles on Integral Projective Curves

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