Noether-Enriques theorem
on canonical curves
A theorem on the projective normality of a canonical curve and on its definability by quadratic equations.
Let be a smooth canonical (non-hyper-elliptic) curve of genus
over an algebraically closed field
and let
be the homogeneous ideal in the ring
defining
in
. The Noether–Enriques theorem (sometimes called the Noether–Enriques–Petri theorem) asserts that:
1) is projectively normal in
;
2) if , then
is a plane curve of degree 4, and if
, then the graded ideal
is generated by the components of degree 2 and 3 (which means that
is the intersection of the quadrics and cubics in
passing through it);
3) is always generated by the components of degree 2, except when a)
is a trigonal curve, that is, has a linear series (system)
, of dimension 1 and degree 3; or b)
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 intersect along a surface
which for a) is non-singular, rational, ruled of degree
in
,
, and the series
cuts out on
a linear system of straight lines on
, and for
a quadric in
(possibly a cone); and for b) is the Veronese surface
in
.
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 is called
-normal if the hypersurfaces of degree
cut out the complete linear system
. Instead of
-normal, linearly normal is used. A curve
is projectively normal if it is
-normal for every
. 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) |
[a2] | E. Arbarello, M. Cornalba, P.A. Griffiths, J.E. Harris, "Geometry of algebraic curves" , 1 , Springer (1984) |
Noether-Enriques theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Noether-Enriques_theorem&oldid=18895