Seshadri constant: Difference between revisions
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It is known that [[Nagata's conjecture on algebraic curves]] is equivalent to the assertion that for more than nine general points, the Seshadri constants of the [[projective plane]] are maximal. There is a general conjecture for [[algebraic surfaces]], the '''[[Nagata–Biran conjecture]]'''. |
It is known that [[Nagata's conjecture on algebraic curves]] is equivalent to the assertion that for more than nine general points, the Seshadri constants of the [[projective plane]] are maximal. There is a general conjecture for [[algebraic surfaces]], the '''[[Nagata–Biran conjecture]]'''. |
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==Definition== |
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Let X be a smooth projective variety, L an ample line bundle on it, x a point of X, <math>{ \mathcal{C}_x }</math> = { all irreducible curves passing through x }. |
Let <math>{ X }</math> be a smooth [[projective variety]], <math>{ L }</math> an [[ample line bundle]] on it, <math>{ x }</math> a point of <math>{ X }</math>, <math>{ \mathcal{C}_x }</math> = { all irreducible curves passing through <math>{ x }</math> }. |
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<math>{ \epsilon ( L, x ) := \underset{ C \in \mathcal{C}_x } { \inf } \frac{ L \cdot C } { |
<math>{ \epsilon ( L, x ) := \underset{ C \in \mathcal{C}_x } { \inf } \frac{ L \cdot C } { \operatorname{mult}_x( C ) } }</math>. |
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Definition. one says that <math>{ \epsilon (L, x) }</math> is the Seshadri constant of L at the point x. |
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Definition: One says that <math>{ \epsilon (L, x) }</math> is the Seshadri constant of <math>{ L }</math> at the point <math>{ x }</math>, a real number. When <math>{ X }</math> is an [[abelian variety]], it can be shown that <math>{ \epsilon (L, x) }</math> is independent of the point chosen, and it is written simply <math>{ \epsilon (L) }</math>. |
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==References== |
==References== |
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*{{ citation | last1 = |
*{{ citation | last1 = Lazarsfeld | first1 = Robert | title = Positivity in Algebraic Geometry I - Classical Setting: Line Bundles and Linear Series | publisher = Springer-Verlag Berlin Heidelberg | year = 2004 | pages = 269–270 }} |
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*{{ citation | last1 = Bauer | first1 = Thomas | last2 = Grimm | first2 = Felix Fritz | last3 = Schmidt | first3 = |
*{{ citation | last1 = Bauer | first1 = Thomas | last2 = Grimm | first2 = Felix Fritz | last3 = Schmidt | first3 = Maximilian | title = On the Integrality of Seshadri Constants of Abelian Surfaces | year = 2018 | arxiv = 1805.05413 }} |
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[[Category:Algebraic varieties]] |
[[Category:Algebraic varieties]] |
Latest revision as of 13:31, 2 April 2023
In algebraic geometry, a Seshadri constant is an invariant of an ample line bundle L at a point P on an algebraic variety. It was introduced by Demailly to measure a certain rate of growth, of the tensor powers of L, in terms of the jets of the sections of the Lk. The object was the study of the Fujita conjecture.
The name is in honour of the Indian mathematician C. S. Seshadri.
It is known that Nagata's conjecture on algebraic curves is equivalent to the assertion that for more than nine general points, the Seshadri constants of the projective plane are maximal. There is a general conjecture for algebraic surfaces, the Nagata–Biran conjecture.
Definition
[edit]Let be a smooth projective variety, an ample line bundle on it, a point of , = { all irreducible curves passing through }.
.
Here, denotes the intersection number of and , measures how many times passing through .
Definition: One says that is the Seshadri constant of at the point , a real number. When is an abelian variety, it can be shown that is independent of the point chosen, and it is written simply .
References
[edit]- Lazarsfeld, Robert (2004), Positivity in Algebraic Geometry I - Classical Setting: Line Bundles and Linear Series, Springer-Verlag Berlin Heidelberg, pp. 269–270
- Bauer, Thomas; Grimm, Felix Fritz; Schmidt, Maximilian (2018), On the Integrality of Seshadri Constants of Abelian Surfaces, arXiv:1805.05413