Concurrent lines on del Pezzo surfaces of degree one

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Concurrent lines on

Del Pezzo surfaces of degree one

Ronald van Luijk Rosa Winter

May 29, 2015 Rio de Janeiro

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Cubic surfaces

LetX ⊂ P3 be a smooth cubic surface over a field k = k. Theorem (Cayley-Salmon).

The surfaceX contains exactly27 lines. Each point lies on at most3 lines.

Proof of second part.

The lines through a pointP lie in the tangent plane H, so in the cubic curveX ∩ H.

Definition.

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Cubic surfaces

Definition.

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Cubic surfaces

Definition.

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Cubic surfaces

Example.

The Fermat surfacew3+ x3+ y3+ z3 = 0 contains the 9lines

w3+ x3 = y3+ z3= 0.

Three of these go through[1 : −1 : 0 : 0]. In total: 27lines and 18 Eckardt points.

Fact. There are45 tritangent planes.

Fact (Hirschfeld ’67).

At most45 Eckardt points (sharp in characteristic 2). At most18 Eckardt points in characteristic not equal to 2.

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Cubic surfaces

Example.

The Fermat surfacew3+ x3+ y3+ z3 = 0 contains the 9lines

w3+ x3 = y3+ z3= 0.

Three of these go through[1 : −1 : 0 : 0]. In total: 27lines and 18 Eckardt points.

Fact. There are45 tritangent planes.

Fact (Hirschfeld ’67).

At most45 Eckardt points (sharp in characteristic 2). At most18 Eckardt points in characteristic not equal to 2.

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Cubic surfaces (another point of view)

Fact. The cubic surfaceX ⊂ P3 is isomorphic to the blow up of P2

in6 points in general position: no 3on a line, no6 on a conic.

Fact. The canonical divisorKX is linearly equivalent with

(−n −1 +d )H = (−3 −1 +3)H = −H for any hyperplane sectionH.

Fact (Riemann-Roch and adjunction). The lines onX correspond with the classesE in Pic X with E2 = −1and H · E = 1.

Fact. Pic X is the orthogonal direct sum ZL ⊕L6i =1ZEi, where

π : X → P2 is the blow up, Lis the pull back of a line, and the Ei

are the exceptional curves. Moreover,L2 = 1,E_i2 = −1,L · Ei = 0,

Ei · Ej = 0 (i 6= j ).

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Cubic surfaces (another point of view)

Fact (Riemann-Roch and adjunction). The lines onX correspond with the classesE in Pic X with E2 _{= −1}_and _{H · E = 1}_.

π : X → P2 is the blow up, Lis the pull back of a line, and the Ei

Ei · Ej = 0 (i 6= j ).

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Cubic surfaces (another point of view)

Fact (Riemann-Roch and adjunction). The lines onX correspond with the classesE in Pic X with E2 _{= −1}_and _{H · E = 1}_.

π : X → P2 _{is the blow up,} _L_{is the pull back of a line, and the} _E i

Ei · Ej = 0 (i 6= j ).

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Cubic surfaces (another point of view)

Corollary. The classesE ∈ Pic X with E2= −1 and−KX · E = 1:

Ei for 1 ≤ i ≤ 6,

L − Ei− Ej for 1 ≤ i < j ≤ 6,

2L + Ei−P Ej for 1 ≤ i ≤ 6,

corresponding to the6 exceptional curves and strict transforms of the15 lines through two points and

the6 conics through five of the six points.

Corollary. The incidence graph on the27 lines is independent ofX. It is thecomplementof the Schl¨afli graph.

Schl¨afli graph. Each line is connected to

10 other lines, partitioned in five disjoint pairs of intersecting lines.

Corollary. (Concurrent lines form a com-plete subgraph, so) each point on at most3

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Cubic surfaces (another point of view)

Ei for 1 ≤ i ≤ 6,

L − Ei− Ej for 1 ≤ i < j ≤ 6,

2L + Ei−P Ej for 1 ≤ i ≤ 6,

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Cubic surfaces (another point of view)

Ei for 1 ≤ i ≤ 6,

L − Ei− Ej for 1 ≤ i < j ≤ 6,

2L + Ei−P Ej for 1 ≤ i ≤ 6,

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Del Pezzo surfaces

Adel Pezzo surfaceover a field k is a geometrically integral, smooth, projective surfaceS overk for which there exists an embeddingi : S ,→ Pn and a positive integerasuch that the multiple−aK_S of a canonical divisor KS is linearly equivalent

to a hyperplane section. Itsdegreeis deg S = K_S2.

Examples.

I A smooth cubic surfaceinP3, with a = 1 and degree 3.

I A double cover of P2 ramified over a smooth quartic curve,

with a = 2 (w.r.t. an embedding inP6) and degree2.

Fact.

Over an algebraically closed field, every del Pezzo surface is isomorphic to

I _P1_{× P}1, with degree 8, or

I _P2 blown up at r ≤ 8points in general position (!), with degree 9 − r (cubic: r = 6).

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Del Pezzo surfaces

General position:

I no3 points on a line,

I no6 points on a conic,

I no8 points on a singular cubic with singularity at one of 8.

The (−1)-curves on the blow-up:

I the exceptional curves, and strict transforms of...

I lines through two (of the) points,

I conics through five points,

I cubics through seven points, singular at one (of them),

I quartics through eight points, singular at three,

I quintics through eight points, singular at six,

I sextics through eight singular points, triple point at one.

r 0 1 2 3 4 5 6 7 8

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Del Pezzo surfaces

General position:

I no3 points on a line,

I no6 points on a conic,

I no8 points on a singular cubic with singularity at one of 8. The (−1)-curves on the blow-up:

I the exceptional curves, and strict transforms of...

I lines through two (of the) points,

I conics through five points,

I cubics through seven points, singular at one (of them),

I quartics through eight points, singular at three,

I quintics through eight points, singular at six,

I sextics through eight singular points, triple point at one.

r 0 1 2 3 4 5 6 7 8

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Del Pezzo surfaces of degree two

These are double covers ofP2, ramified over smooth quartic curve.

The28 bitangents pull back to 56“lines”, that is, (−1)-curves. Each linee intersects itspartnere0 with multiplicity2.

Each other line intersects exactly one ofe ande0 with multiplicity1.

Corollary. Each line intersects27lines with multiplicity 1.

Fact. The subgraph on these27 lines is the graph for cubics !

Reason. True for27 lines intersected with multiplicity0 and there is an automorphism that sends each line to its partner.

Corollary. Each point of a del Pezzo surfaceX of degree2 lies on at most4lines (generalised Eckardt point: not on branch curve). There are at most 56 ·₄₋₁27 ·1

4 = 126generalised Eckardt points.

This upper bound is sharp: w2 = x4+ y4+ z4 overF9.

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Del Pezzo surfaces of degree two

Corollary. Each point of a del Pezzo surfaceX of degree2 lies on at most4lines (generalised Eckardt point: not on branch curve). There are at most 56 ·₄₋₁27 ·1

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Del Pezzo surfaces of degree two

Corollary. Each point of a del Pezzo surfaceX of degree2lies on at most4lines (generalised Eckardt point: not on branch curve). There are at most56 ·₄₋₁27 ·1

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Del Pezzo surfaces: automorphism groups (Manin)

LetX be the blow-up ofP2 in 6 ≤ r ≤ 8points in general position.

LetE be the set of classes corresponding to lines. ThenE lies in the hyperplane in thelatticePic X given by −KX · e = 1.

Aut Pic X ⊃ {σ : σKX = KX} ∼ =

−→ Aut K_X⊥−→ W (E∼= r)

↓∼=

Sym(E ) ⊃ {σ : σ(e) · σ(e0) = e · e0 for all e, e0 ∈ E} =: G_r

whereE6, E7, E8 are the classical root lattices (ofx withx2 = −2).

#G6 = 27· 34· 5 = 51840,

#G7 = 210· 34· 5 · 7 = 2903040,

#G8 = 214· 35· 52· 7 = 696729600.

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Del Pezzo surfaces: automorphism groups (Manin)

LetX be the blow-up ofP2 in 6 ≤ r ≤ 8points in general position.

LetE be the set of classes corresponding to lines. ThenE lies in the hyperplane in thelatticePic X given by −KX · e = 1.

Aut Pic X ⊃ {σ : σKX = KX} ∼ =

−→ Aut K_X⊥−→ W (E∼= r)

↓∼=

Sym(E ) ⊃ {σ : σ(e) · σ(e0) = e · e0 for all e, e0 ∈ E} =: G_r

whereE6, E7, E8 are the classical root lattices (ofx withx2 = −2).

#G6 = 27· 34· 5 = 51840,

#G7 = 210· 34· 5 · 7 = 2903040,

#G8 = 214· 35· 52· 7 = 696729600.

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Del Pezzo surfaces of degree one

Every del Pezzo surfaceX of degree one is the double cover of a coneC ⊂ P3, ramified over a curve B of degree 6: the intersection ofC with a surface of degree 3.

Fact. There are120planes that are tritangent toB and do not contain the vertex of the cone. They pull back to240lines onX. Each intersects itspartnerwith multiplicity3.

Fact. If two partnered lines go through a pointP, then P lies on the ramification curve.

Fact. IfP lies on the ramification curve, then the partner of any line throughP also goes throughP.

Fact. Ife ande0 are partners, thene · f = 2 − e0· f for all linesf. (e + e0∼ −2KX)

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Del Pezzo surfaces of degree one

Fact. There are120planes that are tritangent toB and do not contain the vertex of the cone. They pull back to240lines onX. Each intersects itspartner with multiplicity3.

Fact. IfP lies on the ramification curve, then the partner of any line throughP also goes throughP.

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Del Pezzo surfaces of degree one

Fact. There are120planes that are tritangent toB and do not contain the vertex of the cone. They pull back to240lines onX. Each intersects itspartner with multiplicity3.

Fact. IfP lies on the ramification curve, then the partner of any line throughP also goes through P.

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Del Pezzo surfaces of degree one

e1 1 126 L − e1− e2 60 32 1 0 56 56 e2 -1 3 2 1 0

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Del Pezzo surfaces of degree one

Fact (Manin). The groupG = G8 acts transitively on

U = { (e1, e2, . . . , e8) ∈ E8 : ei · ej = 0 for i 6= j }.

Fact. For everyu = (e1, . . . , e8) ∈ U, we can blow downe1, . . . , e8

and there is a unique` ∈ Pic X such that −K_X = 3` −P

iei.

SinceG acts faithfully on Pic X, it acts freely on U, so |U| = |G |.

This action allows us to find the maximal complete subgraphs inside the graph on all240lines.

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Del Pezzo surfaces of degree one

Fact (Manin). The groupG = G8 acts transitively on

U = { (e1, e2, . . . , e8) ∈ E8 : ei · ej = 0 for i 6= j }.

Fact. For everyu = (e1, . . . , e8) ∈ U, we can blow downe1, . . . , e8

and there is a unique` ∈ Pic X such that −K_X = 3` −P

iei.

SinceG acts faithfully on Pic X, it acts freely on U, so |U| = |G |.

This action allows us to find the maximal complete subgraphs inside the graph on all240lines.

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Lemma. The groupG acts transitively on the sets

V1 = { (e1, e2) : e1· e2 = 0 }

V2 = { (e0, e1, e2) : e0· e1= e0· e2 = 1 and e1· e2= 0 }

V3 = { (e0, e1) : e0· e1 = 1 }.

Proof. For V1: the stabiliser Ge1 is isomorphic toG7 and the

subgraph on{e : e · e1= 0}corresponds with r = 7. We have

#V1 = 240 · 56.

One fiber of the mapV2 → V1 has size72, so all fibers do, so

#V2 = 240 · 56 · 72. For one specificv = (e0, e1, e2) ∈ V2, the set

Wv = {e : e · e0= e · e1 = e · e2 = 0}

has6elements, and the stabiliser Gv injects into Sym(Wv) ∼= S6,

so#Gv ≤ 720. Hence, the orbit has size

#Gv = #G

#Gv

≥ #G

720 = #V2,

so we have equality, so the action is transitive.

TheG-action on the image of the projectionρ : V2→ V3 is

transitive. One fiber has size32, so all non-empty fibers do. Hence,

# im ρ = #V2

32 =

240 · 56 · 72

32 = 240 · 126 = #V3,

so ρ is surjective andG acts transitively on V3.

Corollary. For anyv = (e0, e1, e2) ∈ V2, there is a blow-down

X → P2 such thate1, e2 are exceptional curves above two of the

eight points blown up, ande0 is the strict transform of the line in

P2 through these two points.

Proof. LetE1, . . . , E8 be the exceptional curves onX above the

eight pointsP1, . . . , P8 ∈ P2 that were blown up to define X. Let

E0 be the strict transform of the line throughP1 andP2. Let

g ∈ G ⊂ Sym(E )be an element that sends

(E0, E1, E2) to (e0, e1, e2).

Theng sendsE1, E2, . . . , E8 to elementse1, e2, . . . , e8 that we can

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V1 = { (e1, e2) : e1· e2 = 0 }

V2 = { (e0, e1, e2) : e0· e1= e0· e2 = 1 and e1· e2= 0 }

V3 = { (e0, e1) : e0· e1 = 1 }.

subgraph on{e : e · e1 = 0}corresponds with r = 7. We have

#V1 = 240 · 56.

Wv = {e : e · e0= e · e1 = e · e2 = 0}

#Gv = #G

#Gv

≥ #G

720 = #V2,

# im ρ = #V2

32 =

240 · 56 · 72

32 = 240 · 126 = #V3,

(E0, E1, E2) to (e0, e1, e2).

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V1 = { (e1, e2) : e1· e2 = 0 }

V2 = { (e0, e1, e2) : e0· e1= e0· e2 = 1 and e1· e2= 0 }

V3 = { (e0, e1) : e0· e1 = 1 }.

subgraph on{e : e · e1 = 0}corresponds with r = 7. We have

#V1 = 240 · 56.

Wv = {e : e · e0= e · e1 = e · e2 = 0}

#Gv = #G

#Gv

≥ #G

720 = #V2,

# im ρ = #V2

32 =

240 · 56 · 72

32 = 240 · 126 = #V3,

(E0, E1, E2) to (e0, e1, e2).

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V1 = { (e1, e2) : e1· e2 = 0 }

V2 = { (e0, e1, e2) : e0· e1= e0· e2 = 1 and e1· e2= 0 }

V3 = { (e0, e1) : e0· e1 = 1 }.

Proof.

For V1: the stabiliser Ge1 is isomorphic toG7 and the

#V1 = 240 · 56.

Wv = {e : e · e0= e · e1 = e · e2 = 0}

#Gv = #G

#Gv

≥ #G

720 = #V2,

# im ρ = #V2

32 =

240 · 56 · 72

32 = 240 · 126 = #V3,

(E0, E1, E2) to (e0, e1, e2).

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V1 = { (e1, e2) : e1· e2 = 0 }

V2 = { (e0, e1, e2) : e0· e1= e0· e2 = 1 and e1· e2= 0 }

V3 = { (e0, e1) : e0· e1 = 1 }.

#V1 = 240 · 56.

Wv = {e : e · e0= e · e1 = e · e2 = 0}

#Gv = #G

#Gv

≥ #G

720 = #V2,

# im ρ = #V2

32 =

240 · 56 · 72

32 = 240 · 126 = #V3,

(E0, E1, E2) to (e0, e1, e2).

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V1 = { (e1, e2) : e1· e2 = 0 }

V2 = { (e0, e1, e2) : e0· e1= e0· e2 = 1 and e1· e2= 0 }

V3 = { (e0, e1) : e0· e1 = 1 }.

#V1 = 240 · 56.

Wv = {e : e · e0= e · e1 = e · e2 = 0}

#Gv = #G

#Gv

≥ #G

720 = #V2,

# im ρ = #V2

32 =

240 · 56 · 72

32 = 240 · 126 = #V3,

eight pointsP1, . . . , P8 ∈ P2 that were blown up to defineX. Let

(E0, E1, E2) to (e0, e1, e2).

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Del Pezzo surfaces of degree one (example over F

32

)

The maximal size of a complete subgraph is16. There is an example with16 concurrent lines!

SetF = F2[α] = F2[x ]/(x5+ x2+ 1)andP = [0 : 0 : 1] ∈ P2F and:

Q1= (0 : 1 : 1), Q5 = (1 : 1 : 1),

Q2= (0 : 1 : α19), Q6 = (α4 : α4 : 1),

Q3= (1 : 0 : 1), Q7 = (α24: α25: 1),

Q4= (1 : 0 : α5), Q8 = (α25: α26: 1).

Then these curves go throughP (with1 ≤ i ≤ 4):

I the four lines through Q2i −1 andQ2i,

I the four cubics through allQj with j 6= 2i − 1, singular at Q2i,

I the four cubics through all Qj with j 6= 2i, singular at Q2i −1,

I the four quintics through all Qj, singular when j 6= 2i , 2i − 1.

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Del Pezzo surfaces of degree one (example over F

32

)

The maximal size of a complete subgraph is16. There is an example with16 concurrent lines!

SetF = F2[α] = F2[x ]/(x5+ x2+ 1)andP = [0 : 0 : 1] ∈ P2F and:

Q1 = (0 : 1 : 1), Q5 = (1 : 1 : 1),

Q2 = (0 : 1 : α19), Q6 = (α4 : α4 : 1),

Q3 = (1 : 0 : 1), Q7 = (α24: α25: 1),

Q4 = (1 : 0 : α5), Q8 = (α25: α26: 1).

Then these curves go throughP (with1 ≤ i ≤ 4):

I the four lines through Q2i −1 andQ2i,

I the four cubics through allQj withj 6= 2i − 1, singular at Q2i,

I the four cubics through all Qj with j 6= 2i, singular at Q2i −1,

I the four quintics through all Qj, singular when j 6= 2i , 2i − 1.

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Del Pezzo surfaces of degree one

Question. What about characteristic0 ?

Two cases: the pointP on and off the ramification curve.

Theorem(Winter, vL).

In both cases the number of concurrent lines is at most10.

Facts (arguments similar to before, to minimise computation).

I Any 6partnered pairs forming a complete subgraphare contained in a maximal clique of size 16, andG acts transitively on the sets of 6such pairs.

I Any 11lines without partners forming a complete subgraph is contained in a clique of size 12 without partners, andG acts transitively on the sets of 11 such lines in any such clique.

Corollary. To show that no6 such pairs (or 11such lines) are concurrent, it suffices to pick any description in P2 of 6such pairs

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Del Pezzo surfaces of degree one

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Del Pezzo surfaces of degree one

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Del Pezzo surfaces of degree one

I Any 11lines without partners forming a complete subgraph is contained in a clique of size 12 without partners, andG acts transitively on the sets of 11 such lines in any such clique. Corollary. To show that no6 such pairs (or 11such lines) are concurrent, it suffices to pick any description inP2 of 6such pairs

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Del Pezzo surfaces of degree one

Proposition (case P on the ramification curve) Assume thatchar k 6= 2.

LetQ1, . . . , Q8 be eight points in P2 in general position.

LetLi be the line through Q2i andQ2i −1 for i ∈ {1, 2, 3, 4},

andCi ,j the unique cubic throughQ1, . . . , Qi −1, Qi +1, . . . , Q8 that

is singular inQj.

Assume that the four linesL1, L2, L3 andL4 all intersect in one

pointP. Then the three cubics C7,8,C8,7, andC6,5 do not all go

throughP.

Proof. Gr¨obner bases with a lot of manual help.

Corollary. No six partnered pairs through one point, so no point on ramification curve lies on more than 10lines.

Fact. In each characteristic there is an example of a del Pezzo surfaceX with 10 concurrent lines.

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Del Pezzo surfaces of degree one

is singular inQj.

throughP.

Corollary. No six partnered pairs through one point, so no point on ramification curve lies on more than 10lines.

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Del Pezzo surfaces of degree one

is singular inQj.

throughP.

Corollary. No six partnered pairs through one point, so no point on ramification curve lies on more than10 lines.

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Del Pezzo surfaces of degree one

is singular inQj.

throughP.

Corollary. No six partnered pairs through one point, so no point on ramification curve lies on more than10 lines.

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Del Pezzo surfaces of degree one

Proposition (case P off the ramification curve) Assume thatchar k = 0.

LetQ1, . . . , Q8 be eight points in P2 in general position. Set L1 is the line through Q1 andQ2,

L2 is the line through Q3 andQ4,

C1 is the conic throughQ1, Q3, Q5, Q6, andQ7,

D1 is the quartic through all points, singular at Q1, Q7, andQ8

D4 is the quartic through all points, singular at Q4, Q5, andQ7.

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Corollary.

No point inX off the ramification curve lies on> 10lines.

Sketch of proof of Corollary.

I These 10 curves are a subset of a set of11 (and even 12) curves that form a complete subgraph without any partnered pairs.

I The group G acts transitively on the set of all sets of11 such curves.

I If there are 11 concurrent lines onX, then (as before), there is a blow down X → P2 such that 10 of these curves have images as described in the proposition.

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Corollary.

No point inX off the ramification curve lies on> 10lines.

Sketch of proof of Corollary.

I These 10 curves are a subset of a set of11 (and even 12) curves that form a complete subgraph without any partnered pairs.

I The group G acts transitively on the set of all sets of11 such curves.

I If there are 11 concurrent lines onX, then (as before), there is a blow down X → P2 such that 10 of these curves have images as described in the proposition.

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Sketch of proof of proposition. Define

(P2)9 ⊃ Γ = {(P, Q1, . . . , Q8) : Q1, Q2, . . . , Q8 not in general pos’n},

(P2)9 ⊃ ∆ = {(P, Q₁, . . . , Q8) : the curves in the prop’n containP}.

We will show∆ ⊂ Γ, or equivalently, Z := ∆ ∩(P2₎9_{\ Γ}_{= ∅}_.

The groupPGL3(k) acts on everything. After showing that for

(P, Q1, . . . , Q8) ∈ Z, no three of P, Q1, Q5, Q6 lie on a line, we

may restrict to (P2)9 ⊃ P = (P, Q1, . . . , Q8) : P = [−1 : 0 : 1] Q1 = [1 : 0 : 1] Q5 = [0 : 1 : 1] Q6 = [0 : −1 : 1] Γ0 = Γ ∩ P ∆0 = ∆ ∩ P Z0 = Z ∩ P = ∆0∩ (P \ Γ0)

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Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 P L1 L2 Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 P C1 Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 P C2 Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 P C3 Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 P C4 Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 P L Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 P P∼= (P2)5_{⊃ Y =} {(Q2, Q3, Q4, Q7, Q8) : P ∈ L1, L2, C1, C2, C3} L5 ν λ µ Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 P t Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 P u Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 P

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L ν λ µ t u Q1 Q5 Q6 P P∼= (P2)5_{⊃ Y =} {(Q2, Q3, Q4, Q7, Q8) : P ∈ L1, L2, C1, C2, C3} L5 L1 ν λ µ t u Q1 Q5 Q6 P C1 ν λ µ t u Q1 Q5 Q6 P C2 ν λ µ t u Q1 Q5 Q6 P L2 ν λ µ t u Q1 Q5 Q6 P Q3 Q4 ν λ µ t u Q1 Q5 Q6 P Q7 ν λ µ t u Q1 Q5 Q6 P Q8 C3 ν λ µ t u Q1 Q5 Q6 P Q3 Q7 Q8 Q2 ν λ µ t u Q1 Q5 Q6 P

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P∼= (P2)5⊃ Y = {(Q2, Q3, Q4, Q7, Q8) : P ∈ L1, L2, C1, C2, C3} L9999K L5

The extra requirementP ∈ C4 yields a hypersurface in

L5(λ, µ, ν, t, u) that is a conic bundle overL3(λ, µ, ν)with a section. Hence, it is birational toA4.

InA4, the four conditionsP ∈ Di for 1 ≤ i ≤ 4define a set that is

contained in the contained in the set that describes not being in general position (at this point magma is able to help out). This proves the proposition.

Proof of the theorem (case P off the ramification curve). The corollary already said that> 10concurrent lines is impossible. There is a2-dimensional family of examples with 10 concurrent lines.

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