Brill–Noether theory

For a given genus {{mvar|g}}, the moduli space for curves {{mvar|C}} of genus {{mvar|g}} 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 {{mvar|d}}, as a function of {{mvar|g}}, that must be present on a curve of that genus.

The basic statement can be formulated in terms of the Picard variety {{math|Pic(C)}} of a smooth curve {{mvar|C}}, and the subset of {{math|Pic(C)}} corresponding to divisor classes of divisors {{mvar|D}}, with given values {{mvar|d}} of {{math|deg(D)}} and {{mvar|r}} of {{math|l(D) – 1}} in the notation of the Riemann–Roch theorem. There is a lower bound {{mvar|ρ}} for the dimension {{math|dim(d, r, g)}} of this subscheme in {{math|Pic(C)}}:

:$\backslash dim(d,r,g)\; \backslash geq\; \backslash rho\; =\; g-(r+1)(g-d+r)$

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 {{mvar|C}} and for {{math|d ≥ 1}}, {{math|r ≥ 0}} the basic results about the space {{tmath|G^r_d}} of linear systems on {{mvar|C}} of degree {{mvar|d}} and dimension {{mvar|r}} are as follows.

• George Kempf proved that if {{math|ρ ≥ 0}} then {{tmath|G^r_d}} is not empty, and every component has dimension at least {{mvar|ρ}}.
• William Fulton and Robert Lazarsfeld proved that if {{math|ρ ≥ 1}} then {{tmath|G^r_d}} is connected.
• {{harvtxt|Griffiths|Harris|1980}} showed that if {{mvar|C}} is generic then {{tmath|G^r_d}} is reduced and all components have dimension exactly {{mvar|ρ}} (so in particular {{tmath|G^r_d}} is empty if {{math|ρ < 0}}).
• David Gieseker proved that if {{mvar|C}} is generic then {{tmath|G^r_d}} is smooth. By the connectedness result this implies it is irreducible if {{math|ρ > 0}}.

Other more recent results not necessarily in terms of space {{tmath|G^r_d}} of linear systems are:

• Eric Larson (2017) proved that if {{math|ρ ≥ 0}}, {{math|r ≥ 3}}, and {{math|n ≥ 1}}, the restriction maps $H^0(\backslash mathcal\{O\}\_\{\backslash mathbb\{P\}^r\}(n))\backslash rightarrow\; H^0(\backslash mathcal\{O\}\_\{C\}(n))$ are of maximal rank, also known as the maximal rank conjecture.{{cite arXiv |eprint=1711.04906 |class=math.AG |first=Eric |last=Larson |title=The Maximal Rank Conjecture |date=2018-09-18}}{{Cite web |last=Hartnett |first=Kevin |date=2018-09-05 |title=Tinkertoy Models Produce New Geometric Insights |url=https://www.quantamagazine.org/tinkertoy-models-produce-new-geometric-insights-20180905/ |access-date=2022-08-28 |website=Quanta Magazine |language=en}}

• Eric Larson and Isabel Vogt (2022) proved that if {{math|ρ ≥ 0}} then there is a curve {{mvar|C}} interpolating through {{mvar|n}} general points in {{tmath|\mathbb{P}^r}} if and only if $(r-1)n\; \backslash leq\; (r\; +\; 1)d\; -\; (r-3)(g-1),$ except in 4 exceptional cases: {{math|(d, g, r) ∈ {(5,2,3),(6,4,3),(7,2,5),(10,6,5)}.}}{{cite arXiv |eprint=2201.09445 |class=math.AG |first1=Eric |last1=Larson |first2=Isabel |last2=Vogt |title=Interpolation for Brill--Noether curves |date=2022-05-05}}{{Cite web |date=2022-08-25 |title=Old Problem About Algebraic Curves Falls to Young Mathematicians |url=https://www.quantamagazine.org/old-problem-about-algebraic-curves-falls-to-young-mathematicians-20220825/ |access-date=2022-08-28 |website=Quanta Magazine |language=en}}

• {{cite thesis |type=Master's thesis |title=Algebraic Brill–Noether Theory |first=Andrea |last=Barbon |publisher=Radboud University Nijmegen |date=2014 |url=http://abarbon.com/assets/Andrea%20Barbon%20-%20Algebraic%20Brill-Noether%20Theory.pdf }}
• {{cite book |first1=Enrico |last1=Arbarello |first2=Maurizio |last2=Cornalba |first3=Philip A. |last3=Griffiths |first4=Joe |last4=Harris |chapter=The Basic Results of the Brill-Noether Theory |pages=203–224 |title=Geometry of Algebraic Curves |volume=I |series=Grundlehren der Mathematischen Wissenschaften 267 | year=1985 | isbn=0-387-90997-4 |doi=10.1007/978-1-4757-5323-3_5 }}
• {{Cite journal

| doi=10.1007/BF02104804

| last1=von Brill

| first1=Alexander

| first2=Max

| last2=Noether

| title=Ueber die algebraischen Functionen und ihre Anwendung in der Geometrie

| year=1874

| journal=Mathematische Annalen

| volume=7

| pages=269–316

| jfm=06.0251.01

| url=http://gdz.sub.uni-goettingen.de/en/dms/load/toc/?PPN=PPN235181684_0007&DMDID=dmdlog21

| access-date=2009-08-22

| issue=2

| s2cid=120777748

}}

• {{Cite journal|mr=0563378

|last1=Griffiths|first1= Phillip|last2= Harris|first2= Joseph

|title=On the variety of special linear systems on a general algebraic curve

|journal=Duke Mathematical Journal|volume= 47 |year=1980|issue= 1|pages= 233–272|doi=10.1215/s0012-7094-80-04717-1}}

• {{cite book | author=Eduardo Casas-Alvero| author-link=Eduardo Casas-Alvero | title=Algebraic Curves, the Brill and Noether way | series=Universitext | publisher=Springer | year=2019 | isbn=9783030290153}}
• {{cite book | author=Philip A. Griffiths | author-link=Phillip Griffiths |author2=Joe Harris |author-link2=Joe Harris (mathematician) | title=Principles of Algebraic Geometry | series=Wiley Classics Library | publisher=Wiley Interscience | year=1994 | isbn=978-0-471-05059-9 | page=245}}

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