Editing Brill–Noether theory (section)

==Main theorems of 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 class]]es 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'')}}:

:<math>\dim(d,r,g) \geq \rho = g-(r+1)(g-d+r)</math>

called the '''Brill–Noether number'''. The formula can be memorized via the mnemonic (using our desired <math>h^0(D) = r+1 </math> and Riemann-Roch) 
:<math>g-(r+1)(g-d+r) = g - h^0(D)h^1(D)</math>

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 (mathematician)|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 <math>H^0(\mathcal{O}_{\mathbb{P}^r}(n))\rightarrow H^0(\mathcal{O}_{C}(n))</math> are of maximal rank, also known as the maximal rank conjecture.<ref>{{cite arXiv |eprint=1711.04906 |class=math.AG |first=Eric |last=Larson |title=The Maximal Rank Conjecture |date=2018-09-18}}</ref><ref>{{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}}</ref>

* 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 <math>(r-1)n \leq (r + 1)d - (r-3)(g-1),</math> except in 4 exceptional cases: {{math|(''d'', ''g'', ''r'') ∈ {(5,2,3),(6,4,3),(7,2,5),(10,6,5)}.}}<ref>{{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}}</ref><ref>{{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}}</ref>