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Difference between revisions of "Noether-Enriques theorem"

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====References====
 
====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>
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<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>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P.A. Griffiths,   J.E. Harris,   "Principles of algebraic geometry" , Wiley (Interscience) (1978)</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)</TD></TR></table>
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<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>

Revision as of 21:54, 30 March 2012

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) 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
How to Cite This Entry:
Noether-Enriques theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Noether-Enriques_theorem&oldid=23910
This article was adapted from an original article by V.A. Iskovskikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article