the geometry of divisors with multiplicity on projective curves · 2017. 2. 13. · a tour of...
TRANSCRIPT
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The Geometry of Divisors with Multiplicity onProjective Curves
Mara Ungureanu
Third Meeting of Young Women in Mathematics14 February 2017
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A tour of Brill-Noether theory
Group theory
19th century: group ↔ subset of GLn20th century: abstract groups{
structure and classification of abstract groupsrepresentation theory
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A tour of Brill-Noether theory
Algebraic geometryAlgebraic curves
19th century: irreducible polynomial in two variables
Curve in a higher dimensional projective space ↔ subset ofprojective space defined by polynomial equations
Classification of algebraic curves ↔ classification of all suchsubsets of projective space
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A tour of Brill-Noether theory
Classification of algebraic curves ↔ describing all components ofthe Hilbert scheme whose general point corresponds to an integralcurve
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A tour of Brill-Noether theory
20th century
Abstract curve
Classification of algebraic curves ↔ study of moduli spaces Mg{study of set of all abstract curvesstudy the ways in which a curve can be mapped to Pr
Brill-Noether theory = representation theory for curves
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A tour of Brill-Noether theory
Moduli spaces
Mg = moduli space of curves
Grd(C) = set of all nondegenerate maps C → Pr of degree d
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A tour of Brill-Noether theory
C smooth, genus gf : C → Pr non-degenerateDegree of f = degree of f∗H =: d
f∗H = p1 + 2p2
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A tour of Brill-Noether theory
A grd = (L, V )
I a line bundle L of degree d on C
I an (r + 1)-dimensional vector space V ⊂ H0(L)
f : C → Pr
p 7→ [σ0(p) : . . . : σr(p)]
Grd(C) = space of all grd-s on C
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A tour of Brill-Noether theory
Estimate for dimension of Grd(C)Describe Grd(C) as a determinantal variety over Pic
d(C)
E F
X
φ
Xk(φ) = {p ∈ X | rk(φp) ≤ k}
dimXk(φ) ≥ dimX − (rk(E)− k)(rk(F )− k)
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A tour of Brill-Noether theory
dimGrd(C) ≥ g − (r + 1)(g − d+ r)
Brill-Noether number
ρ(g, r, d) = g − (r + 1)(g − d+ r)
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A tour of Brill-Noether theory
Existence and non-existence results
I ρ ≥ 0⇒ Grd(C) 6= ∅ for any CI ρ < 0⇒ Grd(C) = ∅ for a general C
Results about the geometry of Grd(C) for general C
I dimGrd(C) = ρ
I Grd(C) is smooth
I ρ = 0⇒ C has a finite number of grd-s
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A tour of Brill-Noether theory
Results about the geometry of the grd-s and their correspondingmaps f : C → Pr(both C and the grd-s are general)
I if r ≥ 3, then f is an embeddingI if r = 2, then f maps C birationally to a curve with at most
nodes as singularities
I if r = 1, then f expresses C as a simply branched cover overP1
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Multitangency conditions
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Multitangency conditions
f∗H = p1 + 2p2
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Multitangency conditions and de Jonquières divisors
de Jonquières counts the number of pairs (p1, p2) with
f∗H = p1 + 2p2
for some hyperplane H ⊂ Pr
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Multitangency conditions and de Jonquières divisors
de Jonquières (and Mattuck, Macdonald) count the n-tuples
(p1, . . . , pn)
withf∗H = a1p1 + . . .+ anpn
wherea1 + . . .+ an = d
for some hyperplane H ⊂ Pr
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Multitangency conditions and de Jonquières divisors
P2a1 = a2 = 2⇒ counting bitangent lines
a1 = . . . = am = 2⇒ counting m-tangent linesa1 = 3⇒ counting flex pointsa1 = 4⇒ counting hyperflexes
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Multitangency conditions and de Jonquières divisors
The (virtual) de Jonquières numbers are the coefficients of
t1 · . . . · tn
in(1 + a21t1 + . . .+ a
2ntn)
g(1 + a1t1 + . . .+ antn)d−r−g
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Multitangency conditions and de Jonquières divisors
Space of all divisors of degree d on C
Cd = C × . . .× C︸ ︷︷ ︸d times
/Sd
For examplep1 + 2p2 ∈ C3
We define de Jonquières divisors
p1 + . . .+ pn ∈ Cn
such thatf∗H = a1p1 + . . .+ anpn ∈ Cd
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Multitangency conditions and de Jonquières divisors
grd = (L, V ) with V ⊂ H0(L) and dimV = r + 1
D = p1 + . . .+ pn is de Jonquières divisor
⇔
the map VβD−−→ V |a1p1+...+anpn has kernel
⇔
rk(βD) ≤ dimV − 1 = r
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Multitangency conditions and de Jonquières divisors
V V |a1p1+...+anpn
D = p1 + . . .+ pn
βD
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Multitangency conditions and de Jonquières divisors
V ⊗OCn F
Cn
β
De Jonquières divisors:
DJn = {D ∈ Cn | rk(βD) ≤ r}
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Multitangency conditions and de Jonquières divisors
dimDJn ≥ n− d+ r
Relevant questions
I n− d+ r < 0⇒ non-existence of de Jonquières divisorsI n− d+ r ≥ 0⇒ existence of de Jonquières divisorsI n− d+ r = 0⇒ finite number of de Jonquières divisorsI dimDJn = n− d+ r
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Why do we care?
L = KC
Diaz (’84), Polishchuk (’03), Farkas-Pandharipande (’15),Bainbridge-Chen-Gendron-Grushevsky-Möller (’16), ...
Fix partition µ = (a1, . . . , an) of d = 2g − 2
Hg(µ) = {(C; p1, . . . , pn) |KC admits the de Jonquières divisor a1p1 + . . . anpn}
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Why do we care?
Hg(µ) ⊂Mg,nI Hg(µ) has expected dimensionI compactification H̃g(µ) has expected codimension in Mg,nI fundamental class of H̃g(µ) related to Pixton tautological
class
What if L 6= KC?