Seshadri constant
Encyclopedia
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.
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, a Seshadri constant is an invariant of an ample line bundle
Ample line bundle
In algebraic geometry, a very ample line bundle is one with enough global sections to set up an embedding of its base variety or manifold M into projective space. An ample line bundle is one such that some positive power is very ample...
L at a point P on an algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
. It was introduced by Demailly to measure a certain rate of growth, of the tensor powers of L, in terms of the jets
Jet (mathematics)
In mathematics, the jet is an operation which takes a differentiable function f and produces a polynomial, the truncated Taylor polynomial of f, at each point of its domain...
of the sections
Section (category theory)
In category theory, a branch of mathematics, a section is a right inverse of a morphism. Dually, a retraction is a left inverse...
of the Lk. The object was the study of the Fujita conjecture
Fujita conjecture
In mathematics, Fujita's conjecture is a problem in the theories of algebraic geometry and complex manifolds, unsolved .In complex manifold theory, the conjecture states that for a positive holomorphic line bundle L on a compact complex manifold M with canonical line bundle K, thenis spanned by...
.
The name is in honour of the Indian mathematician C. S. Seshadri
C. S. Seshadri
C.S.Seshadri FRS is an eminent Indian mathematician. He is currently Director-Emeritus of the Chennai Mathematical Institute, and is known for his work in algebraic geometry. The Seshadri constant is named after him. He is also a recipient of the prestigious Padma Bhushan in 2009, the third highest...
.
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
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 that do not intersect...
are maximal. There is a general conjecture for algebraic surfaces, the Nagata–Biran conjecture.