Zeuthen–Segre invariant

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In algebraic geometry, the Zeuthen–Segre invariant I is an invariant of a projective surface found in a complex projective space which was introduced by Zeuthen (1871) and rediscovered by Corrado Segre (1896).

The invariant I is defined to be d – 4g – b if the surface has a pencil of curves, non-singular of genus g except for d curves with 1 ordinary node, and with b base points where the curves are non-singular and transverse.

Alexander (1914) showed that the Zeuthen–Segre invariant I is χ–4, where χ is the topological Euler–Poincaré characteristic introduced by Poincaré (1895), which is equal to the Chern number c2 of the surface.

## References

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* Alexander, J. W. (1914), "Sur les cycles des surfaces algébriques et sur une définition topologique de l'invariant de Zeuthen-Segre", Atti della Accademia Nazionale dei Lincei. Rend. V (2), 23: 55–6255-62&rft.date=1914&rft.aulast=Alexander&rft.aufirst=J.+W.&rfr_id=info:sid/en.wikipedia.org:Zeuthen–Segre+invariant" class="Z3988">
* Baker, Henry Frederick (1933), Principles of geometry. Volume 6. Introduction to the theory of algebraic surfaces and higher loci., Cambridge Library Collection, Cambridge University Press, ISBN 978-1-108-01782-4, MR 2850141 Reprinted 2010
* Fulton, William (1998), Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Berlin, New York: Springer-Verlag, ISBN 978-3-540-62046-4, MR 1644323
* Poincaré, Henri (1895), "Analysis Situs", Journal de l'École Polytechnique, 1: 1–1231-123&rft.date=1895&rft.aulast=Poincaré&rft.aufirst=Henri&rft_id=http://gallica.bnf.fr/ark:/12148/bpt6k4337198/f7.image&rfr_id=info:sid/en.wikipedia.org:Zeuthen–Segre+invariant" class="Z3988">
* Segre, C. (1896), "Intorno ad un carattere delle superficie e delle varietà superiori algebriche.", Atti della Accademia delle Scienze di Torino (in Italian), 31: 485–501485-501&rft.date=1896&rft.aulast=Segre&rft.aufirst=C.&rfr_id=info:sid/en.wikipedia.org:Zeuthen–Segre+invariant" class="Z3988">
* Zeuthen, H. G. (1871), "Études géométriques de quelques-unes des propriétés de deux surfaces dont les points se correspondent un-à-un", Mathematische Annalen, 4, Springer Berlin / Heidelberg: 21–49, doi:10.1007/BF01443296, ISSN 0025-5831, S2CID 12184016921-49&rft.date=1871&rft_id=https://api.semanticscholar.org/CorpusID:121840169#id-name=S2CID&rft.issn=0025-5831&rft_id=info:doi/10.1007/BF01443296&rft.aulast=Zeuthen&rft.aufirst=H.+G.&rfr_id=info:sid/en.wikipedia.org:Zeuthen–Segre+invariant" class="Z3988">

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