Brill–Noether theory explained

In algebraic geometry, Brill–Noether theory, introduced by, is the study of special divisors, certain divisors on a curve that determine more compatible functions than would be predicted. In classical language, special divisors move on the curve in a "larger than expected" linear system of divisors.

Throughout, we consider a projective smooth curve over the complex numbers (or over some other algebraically closed field).

The condition to be a special divisor can be formulated in sheaf cohomology terms, as the non-vanishing of the cohomology of the sheaf of sections of the invertible sheaf or line bundle associated to . This means that, by the Riemann–Roch theorem, the cohomology or space of holomorphic sections is larger than expected.

Alternatively, by Serre duality, the condition is that there exist holomorphic differentials with divisor on the curve.

Main theorems of Brill–Noether theory

For a given genus, the moduli space for curves of genus should contain a dense subset parameterizing those curves with the minimum in the way of special divisors. One goal 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, as a function of, that must be present on a curve of that genus.

The basic statement can be formulated in terms of the Picard variety of a smooth curve, and the subset of corresponding to divisor classes of divisors, with given values of and of in the notation of the Riemann–Roch theorem. There is a lower bound for the dimension of this subscheme in :

\dim(d,r,g) \geq \rho = g - (r +1 )(g - d + r)

called the Brill–Noether number. The formula can be memorized via the mnemonic (using our desired

h ^{0(D)} = r +1 and Riemann-Roch) g - (r +1 )(g - d + r) = g - h ^{0(D)h} ^{1(D)}

For smooth curves and for, the basic results about the space of linear systems on of degree and dimension are as follows.

• George Kempf proved that if then is not empty, and every component has dimension at least .
• William Fulton and Robert Lazarsfeld proved that if then is connected.
• showed that if is generic then is reduced and all components have dimension exactly (so in particular is empty if).
• David Gieseker proved that if is generic then is smooth. By the connectedness result this implies it is irreducible if .

Other more recent results not necessarily in terms of space of linear systems are:

• Eric Larson (2017) proved that if,, and, the restriction maps

H ^{0 ( l{O}} _{P ^{r}} (n)) → H ^{0 ( l{O}} _{C} (n)) are of maximal rank, also known as the maximal rank conjecture.^{[1] [2]}

• Eric Larson and Isabel Vogt (2022) proved that if then there is a curve interpolating through general points in if and only if

(r -1 )n \leq (r + 1 )d - (r -3 )(g -1 ), except in 4 exceptional cases: ^{[3] [4]}

References

• Master's thesis . Algebraic Brill–Noether Theory . Andrea . Barbon . Radboud University Nijmegen . 2014 .
• Book: Enrico . Arbarello . Maurizio . Cornalba . Philip A. . Griffiths . Joe . Harris . The Basic Results of the Brill-Noether Theory . 203–224 . Geometry of Algebraic Curves . I . Grundlehren der Mathematischen Wissenschaften 267 . 1985 . 0-387-90997-4 . 10.1007/978-1-4757-5323-3_5 .
• 10.1007/BF02104804. von Brill. Alexander. Max. Noether. Ueber die algebraischen Functionen und ihre Anwendung in der Geometrie. 1874. Mathematische Annalen. 7. 269–316. 06.0251.01. 2009-08-22. 2. 120777748.
• 0563378. Griffiths. Phillip. Harris. Joseph. On the variety of special linear systems on a general algebraic curve. Duke Mathematical Journal. 47 . 1980. 1. 233–272. 10.1215/s0012-7094-80-04717-1.
• Book: Eduardo Casas-Alvero. Eduardo Casas-Alvero . Algebraic Curves, the Brill and Noether way . Universitext . Springer . 2019 . 9783030290153.
• Book: Philip A. Griffiths . Phillip Griffiths . Joe Harris . Joe Harris (mathematician) . Principles of Algebraic Geometry . Wiley Classics Library . Wiley Interscience . 1994 . 978-0-471-05059-9 . 245.

Notes and References

1. 1711.04906 . math.AG . Eric . Larson . The Maximal Rank Conjecture . 2018-09-18.
2. Web site: Hartnett . Kevin . 2018-09-05 . Tinkertoy Models Produce New Geometric Insights . 2022-08-28 . Quanta Magazine . en.
3. 2201.09445 . math.AG . Eric . Larson . Isabel . Vogt . Interpolation for Brill--Noether curves . 2022-05-05.
4. Web site: 2022-08-25 . Old Problem About Algebraic Curves Falls to Young Mathematicians . 2022-08-28 . Quanta Magazine . en.