Noether-Enriques theorem

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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]).


[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 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.


[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
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