Seshadri constant: Difference between revisions

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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 L^k. 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

Let ${X}$ be a smooth projective variety, ${L}$ an ample line bundle on it, ${x}$ a point of ${X}$ , ${{\mathcal {C}}_{x}}$ = { all irreducible curves passing through ${x}$ }.

${\epsilon (L,x):={\underset {C\in {\mathcal {C}}_{x}}{\inf }}{\frac {L\cdot C}{\operatorname {mult} _{x}(C)}}}$ .

Here, ${L\cdot C}$ denotes the intersection number of ${L}$ and ${C}$ , ${\operatorname {mult} _{x}(C)}$ measures how many times ${C}$ passing through ${x}$ .

Definition: One says that ${\epsilon (L,x)}$ is the Seshadri constant of ${L}$ at the point ${x}$ , a real number. When ${X}$ is an abelian variety, it can be shown that ${\epsilon (L,x)}$ is independent of the point chosen, and it is written simply ${\epsilon (L)}$ .

References

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

@@ Line 5: / Line 5: @@
 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]]'''.
-==Statement==
+==Definition==
-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> }.
 <div style = "text-align:center;">
-<math>{ \epsilon ( L, x ) := \underset{ C \in \mathcal{C}_x } { \inf } \frac{ L \cdot C } { mult_x } }</math>.
+<math>{ \epsilon ( L, x ) := \underset{ C \in \mathcal{C}_x } { \inf } \frac{ L \cdot C } {  \operatorname{mult}_x( C ) } }</math>.
 </div>
-Here, L·C denotes the intersecting number of L and C, <math>{ mult_x( C ) }</math> measures how many times C passing through x.
+Here, <math>{ L \cdot C }</math> denotes the [[intersection number]] of <math>{ L }</math> and <math>{ C }</math>, <math>{ \operatorname{mult}_x( C ) }</math> measures how many times <math>{ C }</math> passing through <math>{ x }</math>.
-Definition. one says that <math>{ \epsilon (L, x) }</math> is the Seshadri constant of L at the point x.
-De facto, when X is an abelian variety, it can be shown that <math>{ \epsilon (L, x) }</math> is independent of the point chosen. Therefore, in the said situation, on can legally drop x, and writes simply <math>{ \epsilon (L) }</math>.
+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>.
 ==References==
-*{{ citation | last1 = Lazasfeld | first1 = Robert| title = Positivity in Algebraic Geometry I: Classical Setting | year = 2004 | pages = 269–270 }}
+*{{ 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 }}
-*{{ citation | last1 = Bauer | first1 = Thomas | last2 = Grimm | first2 = Felix Fritz | last3 = Schmidt | first3 = Maximalian | title = On the Ingegrality of Seshadri Constants of Abelian Surfaces | arxiv = 1805.05413 }}
+*{{ 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 }}
 [[Category:Algebraic varieties]]