# Calculating the Euler Characteristic of the Moduli Space of Curves

Ontology type: schema:ScholarlyArticle      Open Access: True

### Article Info

DATE

2022-02-18

AUTHORS ABSTRACT

The orbifold Euler characteristic of the moduli space ℳg;1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\cal M}_{g;1}}$$\end{document} of genus g smooth curves with one marked point (g ≥ 1) was calculated by Harer and Zagier: χ(ℳg;1)=ζ(1−2g)=−B2g/(2g)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi ({{\cal M}_{g;1}}) = \zeta (1 - 2g) = - {B_{2g}}/(2g)$$\end{document}, where ζ is the Riemann zeta function and B2g is the (2g)th Bernoulli number. We give a shorter proof of this result using only formal power series and classical combinatorics. More... »

PAGES

1-14

TITLE

Combinatorica

ISSUE

N/A

VOLUME

N/A

### Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s00493-021-4688-1

DOI

http://dx.doi.org/10.1007/s00493-021-4688-1

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1145698748

Indexing Status Check whether this publication has been indexed by Scopus and Web Of Science using the SN Indexing Status Tool
Incoming Citations Browse incoming citations for this publication using opencitations.net

JSON-LD is the canonical representation for SciGraph data.

TIP: You can open this SciGraph record using an external JSON-LD service:

[
{
"@context": "https://springernature.github.io/scigraph/jsonld/sgcontext.json",
{
"id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/01",
"inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/",
"name": "Mathematical Sciences",
"type": "DefinedTerm"
},
{
"id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/0101",
"inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/",
"name": "Pure Mathematics",
"type": "DefinedTerm"
}
],
"author": [
{
"affiliation": {
"alternateName": "Univ Lyon, CNRS, Universit\u00e9 Claude Bernard Lyon 1, UMR 5208, Institut Camille Jordan, F-69622, Villeurbanne, France",
"id": "http://www.grid.ac/institutes/grid.7849.2",
"name": [
"Univ Lyon, CNRS, Universit\u00e9 Claude Bernard Lyon 1, UMR 5208, Institut Camille Jordan, F-69622, Villeurbanne, France"
],
"type": "Organization"
},
"familyName": "Lass",
"givenName": "Bodo",
"id": "sg:person.014253662015.66",
"sameAs": [
"https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.014253662015.66"
],
"type": "Person"
}
],
"citation": [
{
"id": "sg:pub.10.1007/s00039-005-0512-0",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1005178176",
"https://doi.org/10.1007/s00039-005-0512-0"
],
"type": "CreativeWork"
},
{
"id": "sg:pub.10.1007/s00026-017-0356-y",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1090378721",
"https://doi.org/10.1007/s00026-017-0356-y"
],
"type": "CreativeWork"
},
{
"id": "sg:pub.10.1088/1126-6708/2009/12/003",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1034597064",
"https://doi.org/10.1088/1126-6708/2009/12/003"
],
"type": "CreativeWork"
},
{
"id": "sg:pub.10.1007/s00026-012-0138-5",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1033825896",
"https://doi.org/10.1007/s00026-012-0138-5"
],
"type": "CreativeWork"
},
{
"id": "sg:pub.10.1007/978-3-662-02414-0",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1027811290",
"https://doi.org/10.1007/978-3-662-02414-0"
],
"type": "CreativeWork"
},
{
"id": "sg:pub.10.1007/978-3-540-38361-1",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1041442733",
"https://doi.org/10.1007/978-3-540-38361-1"
],
"type": "CreativeWork"
},
{
"id": "sg:pub.10.1007/bf02175831",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1008239147",
"https://doi.org/10.1007/bf02175831"
],
"type": "CreativeWork"
},
{
"id": "sg:pub.10.1007/bfb0060799",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1041778859",
"https://doi.org/10.1007/bfb0060799"
],
"type": "CreativeWork"
},
{
"id": "sg:pub.10.1007/bf02102094",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1030083487",
"https://doi.org/10.1007/bf02102094"
],
"type": "CreativeWork"
},
{
"id": "sg:pub.10.1007/bf02099526",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1042090147",
"https://doi.org/10.1007/bf02099526"
],
"type": "CreativeWork"
},
{
"id": "sg:pub.10.1007/bf01390325",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1011455669",
"https://doi.org/10.1007/bf01390325"
],
"type": "CreativeWork"
}
],
"datePublished": "2022-02-18",
"datePublishedReg": "2022-02-18",
"description": "The orbifold Euler characteristic of the moduli space \u2133g;1\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{\\cal M}_{g;1}}$$\\end{document} of genus g smooth curves with one marked point (g \u2265 1) was calculated by Harer and Zagier: \u03c7(\u2133g;1)=\u03b6(1\u22122g)=\u2212B2g/(2g)\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\chi ({{\\cal M}_{g;1}}) = \\zeta (1 - 2g) = - {B_{2g}}/(2g)$$\\end{document}, where \u03b6 is the Riemann zeta function and B2g is the (2g)th Bernoulli number. We give a shorter proof of this result using only formal power series and classical combinatorics.",
"genre": "article",
"id": "sg:pub.10.1007/s00493-021-4688-1",
"isAccessibleForFree": true,
"isPartOf": [
{
"id": "sg:journal.1136493",
"issn": [
"0209-9683",
"1439-6912"
],
"name": "Combinatorica",
"publisher": "Springer Nature",
"type": "Periodical"
}
],
"keywords": [
"moduli space",
"Euler characteristic",
"Riemann zeta function",
"orbifold Euler characteristic",
"formal power series",
"classical combinatorics",
"zeta function",
"power series",
"smooth curve",
"marked point",
"Bernoulli numbers",
"short proof",
"space",
"combinatorics",
"Zagier",
"Harer",
"proof",
"curves",
"point",
"function",
"number",
"characteristics",
"results",
"series",
"B2",
"genus"
],
"name": "Calculating the Euler Characteristic of the Moduli Space of Curves",
"pagination": "1-14",
"productId": [
{
"name": "dimensions_id",
"type": "PropertyValue",
"value": [
"pub.1145698748"
]
},
{
"name": "doi",
"type": "PropertyValue",
"value": [
"10.1007/s00493-021-4688-1"
]
}
],
"sameAs": [
"https://doi.org/10.1007/s00493-021-4688-1",
"https://app.dimensions.ai/details/publication/pub.1145698748"
],
"sdDataset": "articles",
"sdDatePublished": "2022-12-01T06:44",
"sdPublisher": {
"name": "Springer Nature - SN SciGraph project",
"type": "Organization"
},
"sdSource": "s3://com-springernature-scigraph/baseset/20221201/entities/gbq_results/article/article_941.jsonl",
"type": "ScholarlyArticle",
"url": "https://doi.org/10.1007/s00493-021-4688-1"
}
]

HOW TO GET THIS DATA PROGRAMMATICALLY:

JSON-LD is a popular format for linked data which is fully compatible with JSON.

curl -H 'Accept: application/ld+json' 'https://scigraph.springernature.com/pub.10.1007/s00493-021-4688-1'

N-Triples is a line-based linked data format ideal for batch operations.

curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/pub.10.1007/s00493-021-4688-1'

curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/s00493-021-4688-1'

RDF/XML is a standard XML format for linked data.

curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/s00493-021-4688-1'

This table displays all metadata directly associated to this object as RDF triples.

121 TRIPLES      21 PREDICATES      59 URIs      40 LITERALS      4 BLANK NODES

Subject Predicate Object
2 anzsrc-for:0101
3 schema:author Nf5c013716ed54058849af874ed171775
4 schema:citation sg:pub.10.1007/978-3-540-38361-1
5 sg:pub.10.1007/978-3-662-02414-0
6 sg:pub.10.1007/bf01390325
7 sg:pub.10.1007/bf02099526
8 sg:pub.10.1007/bf02102094
9 sg:pub.10.1007/bf02175831
10 sg:pub.10.1007/bfb0060799
11 sg:pub.10.1007/s00026-012-0138-5
12 sg:pub.10.1007/s00026-017-0356-y
13 sg:pub.10.1007/s00039-005-0512-0
14 sg:pub.10.1088/1126-6708/2009/12/003
15 schema:datePublished 2022-02-18
16 schema:datePublishedReg 2022-02-18
17 schema:description The orbifold Euler characteristic of the moduli space ℳg;1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\cal M}_{g;1}}$$\end{document} of genus g smooth curves with one marked point (g ≥ 1) was calculated by Harer and Zagier: χ(ℳg;1)=ζ(1−2g)=−B2g/(2g)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi ({{\cal M}_{g;1}}) = \zeta (1 - 2g) = - {B_{2g}}/(2g)$$\end{document}, where ζ is the Riemann zeta function and B2g is the (2g)th Bernoulli number. We give a shorter proof of this result using only formal power series and classical combinatorics.
18 schema:genre article
19 schema:isAccessibleForFree true
20 schema:isPartOf sg:journal.1136493
21 schema:keywords B2
22 Bernoulli numbers
23 Euler characteristic
24 Harer
25 Riemann zeta function
26 Zagier
27 characteristics
28 classical combinatorics
29 combinatorics
30 curves
31 formal power series
32 function
33 genus
34 marked point
35 moduli space
36 number
37 orbifold Euler characteristic
38 point
39 power series
40 proof
41 results
42 series
43 short proof
44 smooth curve
45 space
46 zeta function
47 schema:name Calculating the Euler Characteristic of the Moduli Space of Curves
48 schema:pagination 1-14
50 N79dbd6a732b84f32a6132e938dedcca8
51 schema:sameAs https://app.dimensions.ai/details/publication/pub.1145698748
52 https://doi.org/10.1007/s00493-021-4688-1
53 schema:sdDatePublished 2022-12-01T06:44
55 schema:sdPublisher Ne93f465230d44a26bfc0da20c1f34d0f
56 schema:url https://doi.org/10.1007/s00493-021-4688-1
58 sgo:sdDataset articles
59 rdf:type schema:ScholarlyArticle
61 schema:value pub.1145698748
62 rdf:type schema:PropertyValue
63 N79dbd6a732b84f32a6132e938dedcca8 schema:name doi
64 schema:value 10.1007/s00493-021-4688-1
65 rdf:type schema:PropertyValue
66 Ne93f465230d44a26bfc0da20c1f34d0f schema:name Springer Nature - SN SciGraph project
67 rdf:type schema:Organization
68 Nf5c013716ed54058849af874ed171775 rdf:first sg:person.014253662015.66
69 rdf:rest rdf:nil
70 anzsrc-for:01 schema:inDefinedTermSet anzsrc-for:
71 schema:name Mathematical Sciences
72 rdf:type schema:DefinedTerm
73 anzsrc-for:0101 schema:inDefinedTermSet anzsrc-for:
74 schema:name Pure Mathematics
75 rdf:type schema:DefinedTerm
76 sg:journal.1136493 schema:issn 0209-9683
77 1439-6912
78 schema:name Combinatorica
79 schema:publisher Springer Nature
80 rdf:type schema:Periodical
81 sg:person.014253662015.66 schema:affiliation grid-institutes:grid.7849.2
82 schema:familyName Lass
83 schema:givenName Bodo
84 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.014253662015.66
85 rdf:type schema:Person
86 sg:pub.10.1007/978-3-540-38361-1 schema:sameAs https://app.dimensions.ai/details/publication/pub.1041442733
87 https://doi.org/10.1007/978-3-540-38361-1
88 rdf:type schema:CreativeWork
89 sg:pub.10.1007/978-3-662-02414-0 schema:sameAs https://app.dimensions.ai/details/publication/pub.1027811290
90 https://doi.org/10.1007/978-3-662-02414-0
91 rdf:type schema:CreativeWork
92 sg:pub.10.1007/bf01390325 schema:sameAs https://app.dimensions.ai/details/publication/pub.1011455669
93 https://doi.org/10.1007/bf01390325
94 rdf:type schema:CreativeWork
95 sg:pub.10.1007/bf02099526 schema:sameAs https://app.dimensions.ai/details/publication/pub.1042090147
96 https://doi.org/10.1007/bf02099526
97 rdf:type schema:CreativeWork
98 sg:pub.10.1007/bf02102094 schema:sameAs https://app.dimensions.ai/details/publication/pub.1030083487
99 https://doi.org/10.1007/bf02102094
100 rdf:type schema:CreativeWork
101 sg:pub.10.1007/bf02175831 schema:sameAs https://app.dimensions.ai/details/publication/pub.1008239147
102 https://doi.org/10.1007/bf02175831
103 rdf:type schema:CreativeWork
104 sg:pub.10.1007/bfb0060799 schema:sameAs https://app.dimensions.ai/details/publication/pub.1041778859
105 https://doi.org/10.1007/bfb0060799
106 rdf:type schema:CreativeWork
107 sg:pub.10.1007/s00026-012-0138-5 schema:sameAs https://app.dimensions.ai/details/publication/pub.1033825896
108 https://doi.org/10.1007/s00026-012-0138-5
109 rdf:type schema:CreativeWork
110 sg:pub.10.1007/s00026-017-0356-y schema:sameAs https://app.dimensions.ai/details/publication/pub.1090378721
111 https://doi.org/10.1007/s00026-017-0356-y
112 rdf:type schema:CreativeWork
113 sg:pub.10.1007/s00039-005-0512-0 schema:sameAs https://app.dimensions.ai/details/publication/pub.1005178176
114 https://doi.org/10.1007/s00039-005-0512-0
115 rdf:type schema:CreativeWork
116 sg:pub.10.1088/1126-6708/2009/12/003 schema:sameAs https://app.dimensions.ai/details/publication/pub.1034597064
117 https://doi.org/10.1088/1126-6708/2009/12/003
118 rdf:type schema:CreativeWork
119 grid-institutes:grid.7849.2 schema:alternateName Univ Lyon, CNRS, Université Claude Bernard Lyon 1, UMR 5208, Institut Camille Jordan, F-69622, Villeurbanne, France
120 schema:name Univ Lyon, CNRS, Université Claude Bernard Lyon 1, UMR 5208, Institut Camille Jordan, F-69622, Villeurbanne, France
121 rdf:type schema:Organization