Castelnuovo curve

In algebraic geometry, a Castelnuovo curve, studied by Castelnuovo (1889), is a curve in projective space Pn of maximal genus g among irreducible non-degenerate curves of given degree d. Castelnuovo showed that the maximal genus is given by the Castelnuovo bound

g≤(n−1)m(m−1)/2+mϵ

where m and ε are the quotient and remainder when dividing d–1 by n–1. Castelnuovo described the curves satisfying this bound, showing in particular that they lie on either a rational normal scroll or on the Veronese surface.

## References

* Castelnuovo, G. (1889), "Ricerche di geometria sulle curve algebriche" (in Italian), Atti Reale Accademia delle Scienze di Torino 24: 346–373
* Griffiths, Phillip; Harris, Joseph (1994), Principles of algebraic geometry, Wiley Classics Library, New York: John Wiley & Sons, ISBN 978-0-471-05059-9

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