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''on canonical curves''
 
''on canonical curves''
  
 
A theorem on the projective normality of a [[Canonical curve|canonical curve]] and on its definability by quadratic equations.
 
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:
+
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) <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" />;
+
1) $  X $
 +
is projectively normal in $  P  ^ {g-} 1 $;
  
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);
+
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) <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;
+
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 <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" />.
+
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 [[#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]]]).
 
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]]]).
Line 17: Line 58:
 
====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>
 
<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====
 
====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.
+
A smooth curve $  C \subset  P  ^ {g-} 1 $
 +
is called $  k $-
 +
normal if the hypersurfaces of degree $  k $
 +
cut out the complete [[Linear system|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. [[#References|[a2]]], p. 140ff and 221ff for more details and results.
  
 
====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) {{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>
 
<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>

Latest revision as of 08:02, 6 June 2020


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
How to Cite This Entry:
Noether-Enriques theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Noether-Enriques_theorem&oldid=47975
This article was adapted from an original article by V.A. Iskovskikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article