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

Comments

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.

References

[a1]	P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) MR0507725 Zbl 0408.14001
[a2]	E. Arbarello, M. Cornalba, P.A. Griffiths, J.E. Harris, "Geometry of algebraic curves" , 1 , Springer (1984) MR2807457 MR0770932 Zbl 05798333 Zbl 0991.14012 Zbl 0559.14017