The intersection of three quadrics



2u¹ 2u² 2u³ 0 0 0 0 0 0 2u⁴ 2u⁵ 2u⁶





and its rank at the point pl(t+)or pl(t−)is equal to one, i.e. lower than on any other point of P-span(_pl(t+)_{, pl}(t−))_.

2.2. The intersection of three quadrics

Consider the intersection

S_x = _P(v_x)∩_P(_Λ²g_x)∩_P(R_x)

= pl(Gr₁(_P(T_xM⊗_C)))∩_P(_Λ²g_x)∩_P(R_x)

of the three quadrics inP(_Λ²T_xM⊗_C). We consider a lineltangent to the quadric P(g_x). By the discussion in the previous section it corresponds to a point in pl(_Gr1(_P(T_xM⊗_C)))_. The condition that the line l is tagent to the quadric P(g_x)is equivalent to the condition that pl(l)∈ _P(_Λ²g_x)_{. So}

π

(id_P(gx)×pl) _P(T)=pl(Gr₁(_P(TxM⊗_C)))∩_P(_Λ²gx). This means that,

S_x =π

(id_P(gx)×pl) _P(T)∩_P(R_x). Therefore S_x must have singularities

Sing(S_x)⊃_Singπ

(_idP(gx)×_pl) _P(T)∩_P(_Rx) = (C₊∩_P(_Rx))∪(C₋∩_P(_Rx))_.

2.2.1 Definition. The variety Sx is called the local invariant of the Riemannian manifold (M,g)at the pointx.

2.2.2Remark. Notice, that if R_x = _κΛ²g_x,κ ∈ _C^∗, the manifold at the point x is a manifold of constant curvature in any two dimensional direction. In such a case, S_x is not defined and we shall not consider such points on M.

In the folowing we assume that the quadric P(R_x)intersects the non-singular points of π

(id_P(_gx)×pl) _P(T)transversally and intersects the singular locus C+∪ C₋ transver-sally as well. It follows by [14], Proposition17.18that S_x is the complete intersection of the quadricsP(vx),P(_Λ²gx),P(Rx).

2.2.3Remark. Recall that two varieties intersect transversally if they intersect transversally at each point of their intersection, i.e. they are smooth at this point and their separate tangent spaces at that point span the tangent space of the ambient variety at that point. In other words if X andY are projective subvarieties of Pⁿ, then Xand Yintersect transversally if at every point u∈X∩Y,TuX⊕TuY=TuPⁿ. Thus transversality depends on the choice of the ambient variety. In particular, transversality always fails whenever two subvarieties are tangent.

Recall that the complete intersection of three quadrics inP⁵is aK3 surface (more details on that can be found in the Appendix A). Thus S_x is a (singular) K3 surface. The quadric P(R_x)interesects the singular locusC₊∪ C₋transversally and each intersectionP(R_x)∩ C₊, P(R_x)∩ C₋, consists of four ordinary double points (the transversal intersection of quadric and conic gives a 0-dimensional variety of degree 4). We wil denote the set of these points by Sing(S_x) ={pl(t¹₊), pl(t²₊), pl(t³₊), pl(t⁴₊), pl(t¹₋), pl(t²₋), pl(t³₋), pl(t⁴₋)}.

Consider now the the algebraic subvariety

S˜_x= π⁻¹(_P(R_x))⊂_P(T_xM⊗_C)×_P(_Λ²T_xM⊗_C).

The next step is to show, that ˜S_x is the resolution of the singular points ofS_x. We consider the map

˜

π: ˜S_x→S_x.

Then ˜Sx is the resolution of the singular points ofSx, if and only if S˜x\π˜⁻¹(Sing(Sx))∼=Sx\Sing(Sx).

By the definiton of ˜π⁻¹ this is indeed an isomorphism.

We would like to compute now ˜π⁻¹(Sing(S_x)), or in other words to find the blow ups of the singular points pl(tⁱ₊), pl(t₋^j ), 1≤ i,j≤4.

2.2.4 Remark. Let’s recall the notion of the blow up of a complex surface at a point. Let q∈U⊂ Xbe an open neighborhood and(x,y)local coordinates such thatq= (_{0, 0})_{in this} coordinate system. Define

U˜ :={((x,y),[z,w])∈U×_P¹ : xw= yz}.

We have then the projection onto the first factor p_U : ˜U → U ((x,y),[z,w]) 7→ (x,y).

If (x,y)6= (0, 0)_{, then} p⁻_U¹((x,y)) = (((x,y)_,[z,w])). Furthermore we have p⁻_U¹(q) ={q} × P¹. This implies that the restriction

p_U : p⁻_U¹(U\ {q})→U\ {q}

is an isomorphism and p⁻_U¹(q) ∼= _P¹ is a curve contracted by p_U to a point. Now let us take the gluing of X and ˜U along X\ {q} and ˜U\ {q} ∼= U\ {q}. In this way we obtain a surface ˜X together with a morphism p : ˜X → X. Notice that p gives an isomoprhism between X\ {q}and ˜X\p⁻¹(q)and contracts the curve P¹ ∼= p⁻¹(q) to the point q. The morphismp : ˜X → X is called the blow-up ofX alongq. The curve p⁻¹(q)∼= P¹ is called exceptional curve or exceptional divisor of the blow-up.

We obtain that

E_i :=π˜⁻¹(pl(tⁱ₊)) ={(tⁱ₊∩t−, pl(tⁱ₊)): t−⊂ F₋} ∼=P¹, (2.25)

F_j :=π˜⁻¹(_pl(_t−^j )) ={(t+∩t₋^j , pl(_t−^j ))_{: t+} ⊂ F+} ∼=_P¹, (2.26) for 1 ≤i,j≤4. Observe that this means, that ˜πis the blow-up ofSx along pl(tⁱ₊), pl(t₋^j )for 1 ≤i,j≤4 and the curves E_i,F_j, for 1≤i,j≤4 are the exceptional divisor of the blow-up.

In other words, ˜πcontracts the curvesE_i to the points pl(tⁱ₊)and the curvesF_jto the points pl(t₋^j )for 1≤i,j≤4.

The branching curveΓx: We will show that the map

˜

τ: ˜S_x→_P(g_x)

is a double branched cover at a generic point, where ˜τis the restriction ofτto ˜Sx. The term

”double branched cover” means, that there exists a closed subset Br of P(g_x), such that ˜τ restricted to ˜Sx\Ram, where Ram := τ˜⁻¹(_Br)is a topological double cover ofP(_gx)\Br.

Points in Br and Ram are called branching points and ramification points respectively. The term ”generic” stands for the fact that as we will see sometimes ˜τ represents a branched double cover followed by a blow-up. Before describing the preimage ˜τ⁻¹(t+∩t−)we would like to be more precise.

The block decomposition of the Riemann curvature operator in dimension four is given by

Rm=





A B

B^t C



, where AandCcorrespond to the operators associated to

W++ ^scal 24 g7g and

W−+ ^scal 24 g7g

respectively andBis the operator associated to the curvature-like tensor 1

2

◦

Ric7g = ¹ 2

Ric−^scal 4 g

7g.

Recall, that in dimension four

whereW+,W−denote the Weyl parts of the curvature andRic the traceless Ricci tensor.^◦ Consider now the block decomposition above and let u = u₁+u₂ ∈ _Λ²₊(T_xM⊗_C)⊕

where W₊ and W₋ correspond to the operators associated to W+ and W− respectively.

Notice, that in the last implication we are using the fact, that π

(id_P(gx)×pl) _P(T) = pl(_Gr1(_P(T_xM⊗_C)))∩_P(_Λ²g_x)_.

By assuming thatµ6=0 and settings= ^λ

µ we obtain a quadratic equation in the variable sgiven by We can consider the previous equation naturally, as an equation that determines Sx. The discriminant of the equation is given by

∆=4

Thus there are three possible cases for the intersection of the quadric and the line.

(i) If∆6=0, then the intersection consists of exactly two distinct points:

• P-span(_pl(t+)_{, pl}(t−))∩_P(_Rx) ={_pl(l)_{, pl}(l⁰)}_{, where pl}(l)_{, pl}(l⁰)6=_pl(t+)_{, pl}(t−)_.

are two distinct points. Both these points are nonsingular points of ˜S_x.

• P-span(pl(tⁱ₊), pl(t−))∩_P(R_x) = {pl(tⁱ₊), pl(l)}, for some i = 1, ..., 4, where pl(l)6=pl(tⁱ₊)_{, pl}(t−)_{. Then}

τ˜⁻¹(tⁱ₊∩t−) ={(t+∩t−, pl(tⁱ₊)), ˜π⁻¹(pl(l))},

are two distinct points. Both these points are nonsingular points of ˜Sx.

• P-span(pl(t+), pl(t₋^j ))∩_P(Rx) = {pl(t^j₋), pl(l)}, for some j = 1, ..., 4, where pl(l)6=pl(t+), pl(t₋^j ). Then

˜

τ⁻¹(t+∩t₋^j ) ={(t+∩t−, pl(t^j₋))_{, ˜}π⁻¹(_pl(l))}_,

are two distinct points. Both these points are nonsingular points of ˜Sx.

• P-span(pl(tⁱ₊), pl(t₋^j ))∩_P(R_x) ={pl(tⁱ₊), pl(t^j₋)}, for somei,j=1, ..., 4. Then

˜

τ⁻¹(tⁱ₊∩t₋^j ) ={(tⁱ₊∩t₋^j , pl(tⁱ₊)),(tⁱ₊∩t₋^j , pl(t₋^j ))}, are two distinct points. Both these points are nonsingular points of ˜S_x.

(ii) If∆=0, but not all coefficients are equal to zero, then the line has exactly one double point in common with the quadric P(R_x), which is possible if and only if the line is tangent to the quadric at that point:

• P-span(pl(t+), pl(t−))∩_P(R_x) = {pl(l)}, where pl(l) 6= pl(t+), pl(t−). This is the case that P-span(_pl(t+)_{, pl}(t−))is tangent to the quadric P(R_x)at the point pl(l). Then

˜

τ⁻¹(t+∩t−) ={π˜⁻¹(_pl(l))}_.

Obviously in this case t+∩t− corresponds to a branching point and ˜π⁻¹(pl(l)) is a ramification point.

(iii) If∆=0 and all coefficients are simultaneously equal to zero, then the line lies entirely inP(R_x):

• P-span(pl(tⁱ₊), pl(t₋^j ))⊂_P(R_x), for somei,j=1, ..., 4. Then

˜

τ⁻¹(tⁱ₊∩t₋^j ) = {π˜⁻¹(_pl(l))_{: pl}(l)∈_P-span(_pl(tⁱ₊)_{, pl}(t^j₋))\ {_pl(tⁱ₊)_{, pl}(t₋^j )}} ∪

∪{(tⁱ₊∩t^j₋, pl(tⁱ₊))} ∪ {(tⁱ₊∩t₋^j , pl(t₋^j ))}=_:P¹

tⁱ+∩t^j−, where P¹

tⁱ₊∩t^j₋

∼= _P-span(pl(tⁱ₊), pl(t^j₋)) ∼= _P¹, since ˜π maps the curve ˜τ⁻¹(tⁱ₊∩ t₋^j )one to one onto the singular line P-span(pl(tⁱ₊), pl(t^j₋)). Heretⁱ₊∩t^j₋ corre-sponds again to a branching point and in this special case the branching curve Γx ⊂_P(g_x)at the pointtⁱ₊∩t^j₋is singular.

Thus the branching curve is described by

Γx = {([a¹,a²]_,[b¹,b²])∈_P(S⁻_x)×_P(S⁺_x)_{: (}²_.²⁸₎ Λ² 2

Λ²

Λ²

The next propositions can be found in Nikulin’s paper [21].

2.2.5Remark. The branching curve will serve as our local invariant in this text. Precisely, we will use this local invariant in oder to obtain a characterization for the singularity models for Type I singularities for four dimensional Ricci flows. The type of the curve is invariant under the choice of basis for T_xM⊗C. For example, as we will see in Chapter 4, the branching curve associated to a point ofS³×R, is a 4-typle diagonal and that ofS²×_S², is a double rectangle.

2.2.6Proposition. Assume that the branching curveΓx has only finite number of singular points.

Thenτ˜ : ˜Sx →_P(gx)is a branched double cover for all points t+∩t− ∈_Γx, except for the singular points, at whichτ˜ is a branched double cover followed by a blow-up.

Recall that for a covering map ˜τ: ˜S_x →_P(g_x), there exists a homeomorpish ˆσ: ˜S_x→S^˜_x, such that ˜τ◦σˆ =τ, that is to say ˆ˜ σis a lift of ˜τ. The map ˆσis called a deck transformation.

2.2.7Proposition. Assume that the branching curveΓx has only finite number of singular points.

Then the deck transformationσˆ of the branched double cover is everywhere defined onS˜_x.

Then there are given on ˜Sxnonsingular rational curves (exceptional curves) ¯E_i := σˆ(E_i)∼= P¹and ¯F_j:=σˆ(F_j)∼=_P¹, where 1≤i,j≤4.

Im Dokument Local invariants of four-dimensional Riemannian manifolds and their application to the Ricci flow (Seite 27-34)