View source for Noether-Enriques theorem

''on canonical curves''

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

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066770/n0667701.png" /> be a smooth canonical (non-hyper-elliptic) curve of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066770/n0667702.png" /> over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066770/n0667703.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066770/n0667704.png" /> be the homogeneous ideal in the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066770/n0667705.png" /> defining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066770/n0667706.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066770/n0667707.png" />. The Noether–Enriques theorem (sometimes called the Noether–Enriques–Petri theorem) asserts that:

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066770/n0667708.png" /> is projectively normal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066770/n0667709.png" />;

2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066770/n06677010.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066770/n06677011.png" /> is a plane curve of degree 4, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066770/n06677012.png" />, then the graded ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066770/n06677013.png" /> is generated by the components of degree 2 and 3 (which means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066770/n06677014.png" /> is the intersection of the quadrics and cubics in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066770/n06677015.png" /> passing through it);

3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066770/n06677016.png" /> is always generated by the components of degree 2, except when a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066770/n06677017.png" /> is a trigonal curve, that is, has a linear series (system) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066770/n06677018.png" />, of dimension 1 and degree 3; or b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066770/n06677019.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066770/n06677020.png" /> intersect along a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066770/n06677021.png" /> which for a) is non-singular, rational, ruled of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066770/n06677022.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066770/n06677023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066770/n06677024.png" />, and the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066770/n06677025.png" /> cuts out on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066770/n06677026.png" /> a linear system of straight lines on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066770/n06677027.png" />, and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066770/n06677028.png" /> a quadric in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066770/n06677029.png" /> (possibly a cone); and for b) is the Veronese surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066770/n06677030.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066770/n06677031.png" />.

This theorem (in a slightly different algebraic formulation) was established by M. Noether in [[#References|[1]]]; a geometric account was given by F. Enriques (on his results see [[#References|[2]]]; a modern account is in [[#References|[3]]], [[#References|[4]]]; a generalization in [[#References|[5]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> M. Noether, "Ueber invariante Darstellung algebraischer Funktionen" ''Math. Ann.'' , '''17''' (1880) pp. 263–284</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> D.W. Babbage, "A note on the quadrics through a canonical curve" ''J. London. Math. Soc.'' , '''14''' : 4 (1939) pp. 310–314</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> B. Saint-Donat, "On Petri's analysis of the linear system of quadrics through a canonical curve" ''Mat. Ann.'' , '''206''' (1973) pp. 157–175</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> 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</TD></TR></table>



====Comments====
A smooth curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066770/n06677032.png" /> is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066770/n06677034.png" />-normal if the hypersurfaces of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066770/n06677035.png" /> cut out the complete [[Linear system|linear system]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066770/n06677036.png" />. Instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066770/n06677037.png" />-normal, linearly normal is used. A curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066770/n06677038.png" /> is projectively normal if it is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066770/n06677039.png" />-normal for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066770/n06677040.png" />. Cf. [[#References|[a2]]], p. 140ff and 221ff for more details and results.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) {{MR|0507725}} {{ZBL|0408.14001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> E. Arbarello, M. Cornalba, P.A. Griffiths, J.E. Harris, "Geometry of algebraic curves" , '''1''' , Springer (1984) {{MR|2807457}} {{MR|0770932}} {{ZBL|05798333}} {{ZBL|0991.14012}} {{ZBL|0559.14017}} </TD></TR></table>