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

• 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: 55–62
• 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: 1–123
• Segre, C. (1896), "Intorno ad un carattere delle superficie e delle varietà superiori algebriche.", Atti della Accademia delle Scienze di Torino (in Italian), 31: 485–501
• Zeuthen, H. G. (1871), "Études géométriques de quelques-unes des propriétés de deux surfaces dont les points se correspondent un-à-un", Mathematische Annalen, Springer Berlin / Heidelberg, 4: 21–49, doi:10.1007/BF01443296, ISSN 0025-5831, S2CID 121840169