Zeuthen–Segre invariant

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 {{harvs|txt|authorlink=Hieronymus Georg Zeuthen|last=Zeuthen|year=1871}} and rediscovered by {{harvs|txt|authorlink=Corrado Segre|first=Corrado|last=Segre|year=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.

{{harvs|txt|authorlink=James Waddell Alexander II|last=Alexander|year=1914}} showed that the Zeuthen–Segre invariant I is χ–4, where χ is the topological Euler–Poincaré characteristic introduced by {{harvs|txt|last=Poincaré|authorlink=Henri Poincaré|year=1895}}, which is equal to the Chern number c₂ of the surface.

References

{{Citation | last1=Alexander | first1=J. W. | author1-link=James Waddell Alexander II | title=Sur les cycles des surfaces algébriques et sur une définition topologique de l'invariant de Zeuthen-Segre | year=1914 | journal=Atti della Accademia Nazionale dei Lincei. Rend. V (2) | volume=23 | pages=55–62}}
{{Citation | last1=Baker | first1=Henry Frederick | authorlink=H. F. Baker | title=Principles of geometry. Volume 6. Introduction to the theory of algebraic surfaces and higher loci. | url=https://archive.org/details/principlesofgeom06bake | publisher=Cambridge University Press | series=Cambridge Library Collection | isbn=978-1-108-01782-4 |mr=2850141| year=1933}} Reprinted 2010
{{Citation | last1=Fulton | first1=William | author1-link=William Fulton (mathematician) | title=Intersection theory | publisher=Springer-Verlag | location=Berlin, New York | series=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] | isbn=978-3-540-62046-4 |mr=1644323 | year=1998 | volume=2}}
{{Citation | last1=Poincaré | first1=Henri | author1-link=Henri Poincaré | title=Analysis Situs | url=https://gallica.bnf.fr/ark:/12148/bpt6k4337198/f7.image | year=1895 | journal=Journal de l'École Polytechnique | volume=1 | pages= 1–123}}
{{Citation | last1=Segre | first1=C. | title=Intorno ad un carattere delle superficie e delle varietà superiori algebriche. | language=Italian | year=1896 | journal=Atti della Accademia delle Scienze di Torino | volume=31 | pages=485–501}}
{{Citation | last1=Zeuthen | first1=H. G. | title=Études géométriques de quelques-unes des propriétés de deux surfaces dont les points se correspondent un-à-un | publisher=Springer Berlin / Heidelberg | year=1871 | journal=Mathematische Annalen | issn=0025-5831 | volume=4 | pages=21–49 | doi=10.1007/BF01443296| s2cid=121840169 }}

Category:Algebraic surfaces