Poincaré series for curve singularities and its behaviour under projections

Moyano‐Fernández, Julio José

doi:10.1016/j.jpaa.2014.09.009

Cited by 9 publications

(6 citation statements)

References 24 publications

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“…The next theorem inverts (3.2.2): we recover H from P . The fact that H can be recovered from P was already proved in [17,Corollary 4.3]. However, we wish to present a more general statement which also clarifies under what condition the inversion works, and which is applied for certain coefficients provided by the Heegaard Floer link homology as well, cf.…”

Section: Poincaré Series and The Alexander Polynomialmentioning

confidence: 84%

Lattice and Heegaard Floer Homologies of Algebraic Links

Gorsky

Némethi

2015

Int Math Res Notices

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We compute the Heegaard Floer link homology of algebraic links in terms of the multivariate Hilbert function of the corresponding plane curve singularities. The main result of the paper identifies four homologies: (a) the Heegaard Floer link homology of the local embedded link, (b) the lattice homology associated with the Hilbert function, (c) the homologies of the projectivized complements of local hyperplane arrangements cut out from the local algebra, and (d) a generalized version of the Orlik-Solomon algebra of these local arrangements. In particular, the Poincaré polynomials of all these homology groups are the same, and we also show that they agree with the coefficients of the motivic Poincaré series of the singularity. 10-01-678, RFBR-13-01-00755, NSh-8462.2010.1, NSF grant DMS-1403560 and the Simons foundation. A. N. is partially supported by OTKA Grants 81203, 100796 and 112735. HEEGAARD FLOER LINK HOMOLOGY2.1. Review of Heegaard Floer link homology. In this subsection we recall certain basic algebraic structures of Heegaard Floer link homology. For more see [15,27,28,29,30,33].

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Section: Poincaré Series and The Alexander Polynomialmentioning

confidence: 84%

Lattice and Heegaard Floer Homologies of Algebraic Links

Gorsky

Némethi

2015

Int Math Res Notices

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“…In the proposition to be proved, we use that expression to state a characterization of likely elements in the support of P(t), extending somehow the formula (5) to the case of several points. This result is the version of [19,Prop. 3.8] for our Poincaré series associated with generalized Weierstraß semigroups; we shall point its proof out for the sake of clarity.…”

Section: Poincaré Series As An Invariant Eq (2) Provides An Expressmentioning

confidence: 70%

Generalized Weierstrass semigroups and their Poincaré series

Moyano‐Fernández

Tenório

Torres

2019

Finite Fields and Their Applications

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We investigate the structure of the generalized Weierstraß semigroups at several points on a curve defined over a finite field. We present a description of these semigroups that enables us to deduce properties concerned with the arithmetical structure of divisors supported on the specified points and their corresponding Riemann-Roch spaces. This characterization allows us to show that the Poincaré series associated with generalized Weierstraß semigroups carry essential information to describe entirely their respective semigroups.

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“…Let S C be the value semigroup with its conductor c ∈ N |I| , and we use the notation 1 J = j 1 j where 1 j is the j-th vector of the canonical basis of Z |I| and, for convenience 1 J = 0 if J = ∅. Then the Hilbert function satisfies the following useful properties: Lemma 2.7 ([11, Lemma 3.4], see also [29]). For any i ∈ I fixed, h C (ℓ + 1 i ) = h(ℓ) + 1 if and only if there exists s ∈ S C with s i = ℓ i and s j ≥ ℓ j for each j.…”

Section: 7mentioning

confidence: 99%

“…Proof of Theorem 4.1. The proof of Theorem 4.1 is based on the inversion formula of Gorsky and Némethi [16], and Moyano-Fernández [29], which recovers the Hilbert function h C from the Poincaré series P C J , thus inverts formula (41) (and by its formal proof, it is valid for any curve germ (C, 0) as well).…”

Section: 3mentioning

confidence: 99%

The delta invariant of curves on rational surfaces II: Poincaré series and topological aspects

Cogolludo-Agustín,

László,

Martín-Morales

et al. 2020

Preprint

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In this article we study abstract and embedded invariants of reduced curve germs via topological techniques. One of the most important numerical analytic invariants of an abstract curve is its delta invariant. Our primary goal is to develop delta invariant formulae for curves embedded in rational singularities in terms of embedded data. The topological machinery not only produces formulae, but it also creates deep connections with the theory of (analytical and topological) multivariable Poincaré series.

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Poincaré series for curve singularities and its behaviour under projections

Cited by 9 publications

References 24 publications

Lattice and Heegaard Floer Homologies of Algebraic Links

Lattice and Heegaard Floer Homologies of Algebraic Links

Generalized Weierstrass semigroups and their Poincaré series

The delta invariant of curves on rational surfaces II: Poincaré series and topological aspects

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