In algebraic geometry, a Seshadri constant is an invariant of an

ample line bundle In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of an ...

''L'' at a point ''P'' on an

algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers ...

. 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 Section, Sectioning or Sectioned may refer to: Arts, entertainment and media * Section (music), a complete, but not independent, musical idea * Section (typography), a subdivision, especially of a chapter, in books and documents ** Section sig ...

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 In mathematics, the Nagata conjecture on curves, named after Masayoshi Nagata, governs the minimal degree required for a plane algebraic curve to pass through a collection of very general points with prescribed multiplicities. History Nagata arri ...

is equivalent to the assertion that for more than nine general points, the Seshadri constants of the

projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that ...

are maximal. There is a general conjecture for

algebraic surfaces In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of di ...

, the

Nagata–Biran conjecture In mathematics, the Nagata–Biran conjecture, named after Masayoshi Nagata and Paul Biran, is a generalisation of Nagata's conjecture on curves to arbitrary polarised surfaces. Statement Let ''X'' be a smooth algebraic surface and ''L'' be an am ...

Definition

Let

be a smooth

projective variety In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables ...

on it,

a point of

= .

Here,

denotes the

intersection number In mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple (more than 2) curves, and accounting properly for ...

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 In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functi ...

, it can be shown that

is independent of the point chosen, and it is written simply

References

* *{{ 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 Algebraic varieties Vector bundles Mathematical constants