# 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 [1]; a geometric account was given by F. Enriques (on his results see [2]; a modern account is in [3], [4]; a generalization in [5]).

#### References

 [1] M. Noether, "Ueber invariante Darstellung algebraischer Funktionen" Math. Ann. , 17 (1880) pp. 263–284 [2] D.W. Babbage, "A note on the quadrics through a canonical curve" J. London. Math. Soc. , 14 : 4 (1939) pp. 310–314 [3] B. Saint-Donat, "On Petri's analysis of the linear system of quadrics through a canonical curve" Mat. Ann. , 206 (1973) pp. 157–175 [4] V.V. Shokurov, "The Noether–Enriques theorem on canonical curves" Math. USSR Sb. , 15 (1971) pp. 361–403 Math. Sb. , 86 : 3 (1971) pp. 367–408 [5] E. Arbarello, E. Sernesi, "Petri's approach to the study of the ideal associated to a special divisor" Invent. Math. , 49 (1978) pp. 99–119

A smooth curve $C \subset P ^ {g-} 1$ is called $k$- normal if the hypersurfaces of degree $k$ cut out the complete linear system $| {\mathcal O} _ {C} ( k) |$. Instead of $1$- normal, linearly normal is used. A curve $C \subset P ^ {g-} 1$ is projectively normal if it is $k$- normal for every $k$. Cf. [a2], p. 140ff and 221ff for more details and results.