In algebraic geometry, a Castelnuovo curve, studied by , is a curve in projective space P
''n'' 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
:
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
In mathematics, a rational normal scroll is a ruled surface of degree ''n'' in projective space of dimension ''n'' + 1. Here "rational" means birational to projective space, "scroll" is an old term for ruled surface, and "normal" refers ...
or on the
Veronese surface.
References
*
*{{Citation , last1=Griffiths , first1=Phillip , author1-link=Phillip Griffiths , last2=Harris , first2=Joseph , author2-link=Joe Harris (mathematician) , title=Principles of algebraic geometry , publisher=
John Wiley & Sons , location=New York , series=Wiley Classics Library , isbn=978-0-471-05059-9 , mr=1288523 , year=1994
Algebraic curves