algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

, 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. Mo ...

. It was introduced by Demailly to measure a certain ''rate of growth'', of the

tensor power In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of being ...

s 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 In mathematics, Fujita's conjecture is a problem in the theories of algebraic geometry and complex manifolds, unsolved . It is named after Takao Fujita, who formulated it in 1985. Statement In complex geometry, the conjecture states that for a posi ...

. The name is in honour of the Indian mathematician

C. S. Seshadri Conjeevaram Srirangachari Seshadri (29 February 1932 – 17 July 2020) was an Indian mathematician. He was the founder and director-emeritus of the Chennai Mathematical Institute, and is known for his work in algebraic geometry. The 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 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 do ...

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.

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 w ...

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 ta ...

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 func ...

, 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