The Geometry of Divisors with Multiplicity onProjective Curves

Mara Ungureanu

Third Meeting of Young Women in Mathematics14 February 2017

A tour of Brill-Noether theory

Group theory

19th century: group ↔ subset of GLn20th century: abstract groups{

structure and classification of abstract groupsrepresentation theory

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

A tour of Brill-Noether theory

Classification of algebraic curves ↔ describing all components ofthe Hilbert scheme whose general point corresponds to an integralcurve

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

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

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

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

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)

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)

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

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

Multitangency conditions

Multitangency conditions

f∗H = p1 + 2p2

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

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

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

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

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

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

Multitangency conditions and de Jonquières divisors

V V |a1p1+...+anpn

D = p1 + . . .+ pn

βD

Multitangency conditions and de Jonquières divisors

V ⊗OCn F

Cn

β

De Jonquières divisors:

DJn = {D ∈ Cn | rk(βD) ≤ r}

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

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}

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?