Special divisor
Encyclopedia
In mathematics
, in the theory of algebraic curve
s, certain divisors on a curve C are particular, in the sense of determining more compatible functions than would be predicted. These are the special divisors. In classical language, they move on the curve in a larger linear system of divisors
.
The condition to be a special divisor D can be formulated in sheaf cohomology
terms, as the non-vanishing of the H1 cohomology of the sheaf of the sections of the invertible sheaf
or line bundle
associated to D. This means that, by the Riemann–Roch theorem
, the H0 cohomology or space of holomorphic sections is larger than expected.
Alternatively, by Serre duality
, the condition is that there exist holomorphic differentials with divisor ≥ −D on the curve.
is the theory of special divisors on generic algebraic curves. It is of interest mainly in the case of genus
In conceptual terms, for g given, the moduli space
for curves of genus g should contain an open, dense subset parametrizing those curves with the minimum in the way of special divisors. The point of the theory is to 'count constants', for those curves: to predict the dimension of the space of special divisors (up to linear equivalence) of a given degree d, as a function of g, that must be present on a curve of that genus.
The theory was stated by the German geometers Alexander von Brill
and Max Noether
in . A rigorous proof was first given by . The basic statement can be formulated in terms of the Picard variety Pic(C), and the subset of Pic(C) corresponding to divisor classes of divisors D, with given values n of deg(D) and r of l(D) in the notation of the Riemann–Roch theorem
. There is a lower bound for the dimension dim(n, r, g) of this subset in Pic(C) (which is a subscheme):
called the Riemann–Brill–Noether theorem. Modern work shows that the bound is tight, and indeed equality holds for curves with generic moduli.
The problem formulation can be carried over into higher dimensions, and there is now a corresponding Brill–Noether theory for some classes of algebraic surface
s.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, in the theory of algebraic curve
Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
s, certain divisors on a curve C are particular, in the sense of determining more compatible functions than would be predicted. These are the special divisors. In classical language, they move on the curve in a larger linear system of divisors
Linear system of divisors
In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear system corresponds to the number of parameters of the family....
.
The condition to be a special divisor D can be formulated in sheaf cohomology
Sheaf cohomology
In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F...
terms, as the non-vanishing of the H1 cohomology of the sheaf of the sections of the invertible sheaf
Invertible sheaf
In mathematics, an invertible sheaf is a coherent sheaf S on a ringed space X, for which there is an inverse T with respect to tensor product of OX-modules. It is the equivalent in algebraic geometry of the topological notion of a line bundle...
or line bundle
Line bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example a curve in the plane having a tangent line at each point determines a varying line: the tangent bundle is a way of organising these...
associated to D. This means that, by the Riemann–Roch theorem
Riemann–Roch theorem
The Riemann–Roch theorem is an important tool in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles...
, the H0 cohomology or space of holomorphic sections is larger than expected.
Alternatively, by Serre duality
Serre duality
In algebraic geometry, a branch of mathematics, Serre duality is a duality present on non-singular projective algebraic varieties V of dimension n . It shows that a cohomology group Hi is the dual space of another one, Hn−i...
, the condition is that there exist holomorphic differentials with divisor ≥ −D on the curve.
Brill–Noether theory
Brill–Noether theory in algebraic geometryAlgebraic 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...
is the theory of special divisors on generic algebraic curves. It is of interest mainly in the case of genus
Genus (mathematics)
In mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...
- g ≥ 3.
In conceptual terms, for g given, the moduli space
Moduli space
In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects...
for curves of genus g should contain an open, dense subset parametrizing those curves with the minimum in the way of special divisors. The point of the theory is to 'count constants', for those curves: to predict the dimension of the space of special divisors (up to linear equivalence) of a given degree d, as a function of g, that must be present on a curve of that genus.
The theory was stated by the German geometers Alexander von Brill
Alexander von Brill
Alexander Wilhelm von Brill was a German mathematician.Born in Darmstadt, Hesse, he attended University of Giessen where he earned his doctorate under supervision of Alfred Clebsch. He held a chair at the University of Tübingen, where Max Planck was among his students.-External links:...
and Max Noether
Max Noether
Max Noether was a German mathematician who worked on algebraic geometry and the theory of algebraic functions. He has been called "one of the finest mathematicians of the nineteenth century".-Biography:...
in . A rigorous proof was first given by . The basic statement can be formulated in terms of the Picard variety Pic(C), and the subset of Pic(C) corresponding to divisor classes of divisors D, with given values n of deg(D) and r of l(D) in the notation of the Riemann–Roch theorem
Riemann–Roch theorem
The Riemann–Roch theorem is an important tool in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles...
. There is a lower bound for the dimension dim(n, r, g) of this subset in Pic(C) (which is a subscheme):
- dim(n, r, g) ≥ r(n − r + 1) − (r − 1)g
called the Riemann–Brill–Noether theorem. Modern work shows that the bound is tight, and indeed equality holds for curves with generic moduli.
The problem formulation can be carried over into higher dimensions, and there is now a corresponding Brill–Noether theory for some classes of algebraic surface
Algebraic surface
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 and so of dimension four as a smooth manifold.The theory of algebraic surfaces is much more complicated than that...
s.