Namespaces
Variants
Actions

Difference between revisions of "Mordellic variety"

From Encyclopedia of Mathematics
Jump to: navigation, search
(cf Kobayashi hyperbolicity)
Line 9: Line 9:
 
A variety is ''algebraically hyperbolic'' if the special set is empty.  Lang conjectured that a variety $X$ is Mordellic if and only if $X$ is algebraically hyperbolic and that this is turn equivalent to $X$ being [[pseudo-canonical variety|pseudo-canonical]].
 
A variety is ''algebraically hyperbolic'' if the special set is empty.  Lang conjectured that a variety $X$ is Mordellic if and only if $X$ is algebraically hyperbolic and that this is turn equivalent to $X$ being [[pseudo-canonical variety|pseudo-canonical]].
  
For a complex algebraic variety $X$ we similarly define the ''analytic special'' or ''exceptional set'' as the Zariski closure of the union of images of non-trivial [[holomorphic map]]s from $\mathbb{C}$ to $X$.  Brody's definition of a hyperbolic variety is that there are no such maps.  Again, Lang conjectured that a hyperbolic variety is Mordellic and more generally that the complement of the analytic special set is Mordellic.  
+
For a complex algebraic variety $X$ we similarly define the ''analytic special'' or ''exceptional set'' as the Zariski closure of the union of images of non-trivial [[holomorphic map]]s from $\mathbb{C}$ to $X$.  Brody's definition of a hyperbolic variety is that there are no such maps: cf [[Kobayashi hyperbolicity]].  Again, Lang conjectured that a hyperbolic variety is Mordellic and more generally that the complement of the analytic special set is Mordellic.  
  
 
====References====
 
====References====
 
* Lang, Serge (1986). "Hyperbolic and Diophantine analysis". Bulletin of the American Mathematical Society 14 (2): 159–205. {{DOI|10.1090/s0273-0979-1986-15426-1}} {{ZBL|0602.14019}}.
 
* Lang, Serge (1986). "Hyperbolic and Diophantine analysis". Bulletin of the American Mathematical Society 14 (2): 159–205. {{DOI|10.1090/s0273-0979-1986-15426-1}} {{ZBL|0602.14019}}.
 
* Lang, Serge (1997). Survey of Diophantine Geometry. Springer-Verlag. ISBN 3-540-61223-8
 
* Lang, Serge (1997). Survey of Diophantine Geometry. Springer-Verlag. ISBN 3-540-61223-8

Revision as of 09:29, 21 December 2014

2020 Mathematics Subject Classification: Primary: 11G [MSN][ZBL]

An algebraic variety which has only finitely many points in any finitely generated field. The terminology was introduced by Serge Lang to enunciate a range of conjectures linking the geometry of varieties to their Diophantine properties.

Formally, let $X$ be a variety defined over an algebraically closed field of characteristic zero: hence $X$ is defined over a finitely generated field $E$. If the set of points $X(F)$ is finite for any finitely generated field extension $F/E$, then $X$ is Mordellic.

The special set for a projective variety $V$ is the Zariski closure of the union of the images of all non-trivial maps from algebraic groups into $V$. Lang conjectured that the complement of the special set is Mordellic.

A variety is algebraically hyperbolic if the special set is empty. Lang conjectured that a variety $X$ is Mordellic if and only if $X$ is algebraically hyperbolic and that this is turn equivalent to $X$ being pseudo-canonical.

For a complex algebraic variety $X$ we similarly define the analytic special or exceptional set as the Zariski closure of the union of images of non-trivial holomorphic maps from $\mathbb{C}$ to $X$. Brody's definition of a hyperbolic variety is that there are no such maps: cf Kobayashi hyperbolicity. Again, Lang conjectured that a hyperbolic variety is Mordellic and more generally that the complement of the analytic special set is Mordellic.

References

  • Lang, Serge (1986). "Hyperbolic and Diophantine analysis". Bulletin of the American Mathematical Society 14 (2): 159–205. DOI 10.1090/s0273-0979-1986-15426-1 Zbl 0602.14019.
  • Lang, Serge (1997). Survey of Diophantine Geometry. Springer-Verlag. ISBN 3-540-61223-8
How to Cite This Entry:
Mordellic variety. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mordellic_variety&oldid=35767