# Noether-Enriques theorem

on canonical curves

A theorem on the projective normality of a canonical curve and on its definability by quadratic equations.

Let $X \subset P ^ {g-} 1$ be a smooth canonical (non-hyper-elliptic) curve of genus $g \geq 3$ over an algebraically closed field $k$ and let $I _ {X}$ be the homogeneous ideal in the ring $k [ x _ {0} \dots x _ {g-} 1 ]$ defining $X$ in $P ^ {g-} 1$. The Noether–Enriques theorem (sometimes called the Noether–Enriques–Petri theorem) asserts that:

1) $X$ is projectively normal in $P ^ {g-} 1$;

2) if $g = 3$, then $X$ is a plane curve of degree 4, and if $g \geq 4$, then the graded ideal $I _ {X}$ is generated by the components of degree 2 and 3 (which means that $X$ is the intersection of the quadrics and cubics in $P ^ {g-} 1$ passing through it);

3) $I _ {X}$ is always generated by the components of degree 2, except when a) $X$ is a trigonal curve, that is, has a linear series (system) $g _ {3} ^ {1}$, of dimension 1 and degree 3; or b) $X$ is of genus 6 and is isomorphic to a plane curve of degree 5;

4) in the exceptional cases a) and b) the quadrics passing through $X$ intersect along a surface $F$ which for a) is non-singular, rational, ruled of degree $g - 2$ in $P ^ {g-} 1$, $g \geq 5$, and the series $g _ {3} ^ {1}$ cuts out on $X$ a linear system of straight lines on $F$, and for $g = 4$ a quadric in $P ^ {3}$( possibly a cone); and for b) is the Veronese surface $V _ {4}$ in $P ^ {5}$.

This theorem (in a slightly different algebraic formulation) was established by M. Noether in ; a geometric account was given by F. Enriques (on his results see ; a modern account is in , ; a generalization in ).

How to Cite This Entry:
Noether-Enriques theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Noether-Enriques_theorem&oldid=47975
This article was adapted from an original article by V.A. Iskovskikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article