Longitud de entrenudos

We describe the double coverπ:S →T using equations in weighted projective coordinates, then by looking at the morphism defined by the bi-anticanonical bundle onT, we have a better understanding of reducible members of the linear systemπ∗| −2KT|, which encode the singularities ofPto be discussed in the next section.

Lemma 6.2.1. Letπ:S →S/τ=T , thenπ∗(−KT)maps S to_P2as a double cover branched along a sextic curve s6inP2

x6₂+x4₂f2+x22f4+ f6=0 where fi is degree i polynomial inC[x0,x1], and S is the zero locus of

y2= x6₂+x4₂f2+x22f4+ f6

in the weighted projective space_P(3,1,1,1)under coordinates[y:x0: x1: x2]. The involution on S induced by this double cover is i:y7→ −y.

The involution on S associated to the double cover to T isτ:x27→ −x2. Thus T can be represented in

P(3,1,1,2)by

y2= x₂3+x2₂f2+x2f4+ f6

Proof. The map ontoP2is inspired from the results on the relative Prym of degree one del Pezzo surface

Meanwhile, the linear system | −2KT| is not very ample but base-point free [KSC], so consider the following

Lemma 6.2.2. Letϕ−2KT be the regular morphism given by this linear system| −2KT|. It is a double cover of a quadric coneΛinP3branched over P∪δ, where P=[1 : 0 : 0 : 0]is the apex of the cone andδis a

sextic curve as the intersection of the cone with a cubic surface c₃in_P3.

ϕ−2KT :T 2:1

−−→Λ,P∪δ

Proof. This essentially comes from examining the anti-canonical ring of T and is mentioned in [DPT]

[A].

Remark 6.2.3. Dolgachev had a slightly different description in [DOL]. The map factors through the birational map to the anti-canonical model of T , and is a double cover of the quadric cone only branched over a sextic curve. This is because in [DOL] T is taken to be a weak del Pezzo surface which has A₁ singularities and they are mapped to the apex of the cone.

Consider a generic hyperplane section inP3, the ambient space of the quadric coneΛ, its pull-back onto

T underϕ−2KT is in| −2KT|.In addition, dimension of the linear system| −2KT|is 3 according to previous lemma 5.3.4. This implies every curve in| −2KT|is the pull-back of a hyperplane intersection of the quadric coneΛ. Denote byH the family of hyperplane sections ofΛ, we extend the surface map to a tower of double covers C →π∗| −2KT| S C0 → | −2KT| T, ∆∈ | −2KT| H →(P3)∨ Λ⊂P3, Λ∩c3=δ π ϕ−2_KT (6.1)

where on the left hand side, each space is a 3-dimensional linear system on the corresponding surface, and the vertical arrows represent double cover of curves. A generic hyperplane section of the quadratic cone is H∩Λ'_P1, its pull-back underϕ−2KT is a smooth integral curveC0in| −2KT|, andπ−1(C0)=Cis smooth integral inπ∗| −2KT|.

the relative Prym as fibered over the space of hyperplanes inP3

Pv,C → |C0|=|H|=(P3)∨

One can check Riemann-Roch on those double cover of curves. Because C0 intersects with ∆ at π∗_(−2K

T).(−2KT)=4 points,Cis a double cover ofC0branched at 4 points. Similarly, hyperplaneHinP3

intersects withδin 6 points, soC0is a double cover of a rational curveH∩Λbranched at 6 points.

Additionally, we show how to relate the surface maps mentioned so far in one commutative square diagram. We recall a classical result of the projective model of K3 surface.

Lemma 6.2.4. Let L be an ample line bundle on K3 surface S and L' OS(C),then either L is very ample, i.e. the morphismϕLis an embedding with image being degree2g(C)−2in_Pg(C)and all curves in|L|are non-hyperelliptic, or in the the following casesϕLis degree 2 and image of S is of degree g(C)−1

1. L2 =2

2. L2 =8and L=2D for some divisor D, D2 =2 3. L.D=2and D2 =0for some divisor D and thus all curves in|L|are hyperelliptic.

Proof. See [DEB] or [SD].

In our caseL=π∗(−2KT), it lies in the case (2) andϕπ∗_(−2KT)is a degree 2 map. We decomposeϕ_π∗_(−2KT)

into v2◦ϕπ∗_(−KT), where v₂ is the Veronese embedding of degree 2. From previous argument, ϕ_π∗_(−KT)

representsS as a double cover ofP2branched along a sextic s6, andv2 embeds its image inP2toP5as a

quartic surface, ϕπ∗_(−2KT) :S 2:1 −−→_P2,s6 v2 −→V4⊂P5

Let us denote byithe involution onS induced byϕπ∗_(−2K

T).

Notice the pull-back of a conic in P2 is a curve in |π∗(−2KT)|, but dimension of the linear system |π∗_(−2K

T)| is 5, equal to the dimension of conics in P2. This implies that any curve in |π∗(−2KT)| is hyperelliptic.

Furthermore, this mapϕπ∗_(−2KT)onS is compatible with the surface maps onT mentioned before in the

following way.

Theorem 6.2.5. There is a commutative diagram

ϕπ∗_(−2KT) : S _P2 V₄⊂_P5 ϕ−2KT : T P(1,1,2) Λ⊂P 3 2:1 2:1 v2 2:1 2:1 2:1 i

on the second row, the first map is a 2 to 1 projection

[y:x₀:x₁ :x₂]7→[x0 :x1:x2]

followed by embedding

[x0:x1 :x2]7→[x₀2 :x0x1:x2₁:x2]

Proof. The map on the second row can be found in [CO]. We can define the second and third vertical maps in a desired way to ensure commutativity, i.e.

P2→P(1,1,2)

[x0,x1,x2]7→[x0:x1:x2₂]

and this induces the vertical map on the right.