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

${\displaystyle g\leq (n-1)m(m-1)/2+m\epsilon }$

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", Atti Reale Accademia delle Scienze di Torino (in Italian), 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, MR 1288523