deforma-tion equivalence of their Fano schemes ([BD85]). Then the Kodaira-Spencer theorem ([Ara12, p. 300]) shows that this is also the diamond of the Fano
scheme of any smooth cubic fourfold.
Remark 3.2.2. We see thatH2,0(F(X)) is one-dimensional, spanned by a non-degenerate formω. It follows thatF(X) is ahyperkähler variety. To see that ω is non-degenerate, note that by construction, it is a quotient of the form p∗1(ωS)⊗p∗2(ωS), whereωS is the symplectic form on the K3 surfaceS. Remark 3.2.3. This also gives the announced (Remark 2.1.3) example of a Fano scheme which is not a Fano variety. Indeed, hadF(X) been Fano, its anticanonical bundleKF∨ would be ample by definition. But from the short exact sequence
0→TF →TG|F →NF /G'Sym3U∨|F →0
we findKF =−c1(TF) =c1(Sym3U∨|F)−c1(TG). By [3264, p. 183]. c1(TG) = 6h. We know that the total Chern class of U is c(U) = 1−σ1+σ1,1, so an argument with the splitting principle as in Theorem 2.2.1 shows thatc1(NF /G) = 6h. Hence the canonical divisor onF is trivial, and especially not antiample.
However, if we repeat the argument for higher-dimensional cubic hypersur-faces, the only difference is in the first Chern class ofTG, which equals (n+ 2)h for ann-dimensional cubic.
We will need some more information on the Fano scheme as a subvariety of the Grassmannian. Hence we prove
Proposition 3.2.4. The class ofF(X)inCH•(G(2,6))is 27σ2,2+ 18σ3,1(by [3264]), and is a variety of dimension 4 and degree 108 under the Plücker embedding.
Proof. This comes down to (Remark 2.2.2) simplifying 9σ1,1(2σ21+σ1,1 and taking the intersection product of this withσ41. This is possible by hand, but we are lazy and ask Macaulay2 to do it for us, see Appendix A.1.
3.3 Unirationality
We give a construction of unirationality for cubic hypersurfaces, and discuss how subvarieties of a cubic fourfold induce other rational parametrizations.
Let us also note that any cubic hypersurface of dimension ≥ 2 has a unirational degree-2 parametrization. This is the result
Proposition 3.3.1. If X is a cubic hypersurface of dimensionn containing a line`, there is a degree-2-rational mapρ:Pn−199KX.
There are many ways of showing this, see for instance [Bea+16, p. 38] for another idea. We adapt that of [CG72].
Proof. Consider the projective bundleB=P(TX|`). A point in this bundle is (p∈L|p∈X, l⊂TpX). Then this is aPn−1-bundle, so there is a rational map P(TX|`)99K∼ Pn−1×P1. A general lineLtangent toX at a pointpwill have a residual intersection withX, sayq.
21
3. Cubic fourfolds and their Fano schemes
Then a rational map φ:B 99KX is given by (p∈L)7→q. Let nowx∈X be a general point, and letK be the planex, `. Thenφ−1(x) is the set of points (p∈L)∈B wherex∈Landpis a singular point ofK.X. SinceK∩X is the
union of`and a quadric curve,φis 2:1.
It is also possible to construct unirational parametrizations of X in another way. Consider a cubicX containing a smooth rational surfaceS of degreed and with sectional genus (, i.e., the genus of a general hyperplane intersection ofS)g. Then we have:
Lemma 3.3.2. Under these conditions, there is a unirational parametrization P4→X of degree
k= d(d−2)
2 + (2−2g)−1 2(S, S) as long ask >0.
Proof. We follow [Bea+16]. The idea is to use the mapS[2]→X defined by taking p+q ∈S[2] to the residual intersection of the line p, q withX. The degree of this map will then equal the number of secants ofS passing through a general point onX. But the number of secants passing through a point ofX is the same number as the number of secants ofS passing through any point in P5.
This number is computable through the double point formula of Fulton.
Suppose thatx∈X is a sufficiently general point, so that the image Seof S under projection fromxonly has isolated singularities. Call the projectionπx. Then the number of secants passing throughxequals the numberδof double points ofSe. The double point formula ([Ful98, p. 166]) gives an equality
2δ=πx∗πx∗[S]−(c(TP4)c(TS)−1)2.
If we lethbe the class of a hyperplane intersection withS, this simplifies, using the self-intersection formula, to1
2δ=S.eSe−(c2TP4|S−c1(TS)c1(TP4) + (c1TS)2−c2TS)
=d2−(6h2+ 3hKS+KS2−χ(S) + 4h2+ 2hKS+ 4h2+ 2hKS)
=d2−S.S−2h(2h+KS)
=d2−S.S−2(2g−2).
Here, we used the fact thatS.S= 6h2+ 3hKS+KS2−χ(S) - we will show this in detail in Section 8.2.
1This formula appears in Hassett ([Bea+16, p. 59]), but withc1TS instead of (c1TS)2, which must be a misprint.
CHAPTER 4
Connections between a cubic fourfold and its variety of lines
We investigate how the geometry of a cubic fourfold influences its variety of lines, and vice versa. Throughout this chapter, X is a cubic fourfold with F(X) =F1(X) its variety of lines.
4.1 The incidence correspondence
We will refer throughout to the following diagram of the incidence correspon-dence:
Σ
q
""
p
X F(X)
(4.1)
Here Σ =PU |F(X), or set-theoretically
Σ ={p∈l|lis a line in X} ⊂X×F(X).
It follows thatpis a P1-bundle. It is also the case thatq is induced by the sheafOΣ(1) ([3264, p. 336]).
Now we define, as in [BD85]:
Definition 4.1.1. TheAbel-Jacobi map is the map α:=p∗q∗: H4(X,Z)→H2(F(X),Z).
We will occasionally think of it as a map on the corresponding Chow groups, and we extend the notationαto implyp∗q∗in other dimensions as well. We will later discuss its properties in detail, but first, we have:
Lemma 4.1.2. LethX, hF(X) be the Chow-theoretic or cohomological class of hyperplane intersections ofX, F(X)respectively. Then we have thatα(h2X) = hF(X)=σ1, andα(h3X) =σ2
Proof. We have thatσ1is the class of lines intersecting a codimension-2 subspace ofP5, andσ2that of lines intersecting a codimension-3 subspace. But these are
exactlyα(hiX), i= 2,3.
4. Connections between a cubic fourfold and its variety of lines
Related to this, the projection formula (Equation (B.1)) will be useful for intersection computations. Suppose that a ∈ CH1(F(X),Z), p∗q∗(b) ∈ CH3(F(X),Z). Then the projection formula ([Har77, App. A]) tells us that
a.(p∗q∗(b)) =p∗(p∗a.q∗b) =p∗(q∗(q∗p∗a.b))
and if we are only interested in the degree of the intersection (generically the number of points), this equalsq∗p∗a.b.
Lemma 4.1.3. The lines forming a rational curve C ⊂ F(X) sweep out a rational surface inX, of the same degree asC.
Proof. LetCbe such a curve. The incidence variety Σ is aP1-bundle overF(X), and so p−1(C) is a rational surface in Σ. By Castelnuovo (Theorem 4.5.2), q(p−1(C)) will also be rational. (Note that this argument also shows that a general point inq(p−1(C)) will lie on one line ofC).
For the degree, consider the intersection with σ1. Using the projection for-mula (Equation (B.1)), this degree is given byd= ([C].σ1) = (q∗[C].p∗(h2X)) =
(p∗q∗[C].h2X).
A closer look at the incidence correspondence We start somewhat more generally.
Lemma 4.1.4. Consider an incidence correspondence as described, withX a hypersurface inPn of degree d. The fibre of pover any point is isomorphic to an intersection of hypersurfaces of degrees(2,3, . . . , d)inPn−1.
[Mbo17]. Let x ∈ X ⊂ Pn+1, and consider the projectivized tangent space PTPn+1,x. LetPbe a hyperplane not containingx. We construct an isomorphism PTPn+1,x → P, by mapping a tangent direction [v] to the intersection of the line determined byxand [v] withP. We may without loss of generality assume x= (1 : 0 : · · · : 0), P = Z(x0). Letf be a polynomial defining X. Since
wherefi is a homogeneous, (possibly zero) polynomial of degree i. Given a line`as above, let (y0, . . . , yn+1) be a point on it. Letλ, µbe the linear forms corresponding toxand this point respectively. Then any point on`is of the form (λ, µy1, . . . , µyn+1). Substituting this in the equation for f, we obtain differential atx, and vanishes along a hyperplane. The other equations are of
degrees 2,3, . . . , d.
Let us note that the fibre need not be a complete intersection of thefi. This can lead to fibres ofpwith greater dimension than for the general point.
Thus we temporarily define:
4.1. The incidence correspondence
Definition 4.1.5. A pointx∈X such thatp−1(x) is two-dimensional, is an Eckardt point
We will investigate such points fully in Chapter 6.
We will now try to describe what sort of subvarieties of X the lines in a curve or surface ofF(X) sweep out. First, we need to compute a Segre class:
Lemma 4.1.6. LetU be the universal subbundle on the grassmannianG(2, n).
Then the Segre class s(U)is given by 1 +σ1+σ2+σ3+σ4 .
Proof. Sincec(U) = 1−σ1+σ1,1, it is simple to check thats(U)c(U) = 1.
Lemma 4.1.7. LetC∈CH1(X)represent a general curve of degreed. Then the degree ofα([C])∈CH2(F(X)) is21d. Let S∈CH2(X)represent a general surface of degreed. Then the degree ofα([S])∈CH3(F(X))is36d.
Proof. As the intersection ofX with a general plane is a degree 3- curve, we should work inH•(X,Z)3 (that is, we invert 3). Here the class of a line onX will be i∗(h33). A curve of degreedon X has the same degree as d3h3. Now take the diagram Equation (4.1) and note that sincep:P →X is determined by OP(1), we havep∗h3=O(1)3 Now we are ready: We have that
The surface is handled similarly; the degree will be d
3σ1(U).[F(X)].σ31= d
3degF(X) = 36d
Here we used the fact that deg(σ2.σ21)|F(X)= 45, which is a simple Schubert cycle computation when we know the class of [F1(X)] from Proposition 3.2.4.
Of course, a degree d-surface with a d−1 uple point P is rational by projection fromP. However, it follows from Lemma 4.1.7 that such surfaces cannot arise as the surfaceS of lines onX intersecting a nonsingular curveC. Ar-uple point onS then arises as the image of a line onX intersectingC inr points. But if a line intersectsC inrpoints, C must be of degree at leastr. But thenS is of degree at leastr21r.
Let us consider instead a plane P ∈P5intersectingX singularly at a point P - this will create a singular degree 3-curve C onX. SinceCis planar, it will be either a nodal or a cuspidal cubic - both are of course rational by projection from the singularity.
We investigate the inverse imagep−1(C).
(This uses the same technique as in [KUT17].)
Proposition 4.1.8. Let C be a possibly singular irreducible curve of degree at least 2in P5. Then α(C), the variety in F(X) of lines intersecting C, is singular at and only at
25
4. Connections between a cubic fourfold and its variety of lines
• lines intersecting C in a singular point of C
• secant and tangent lines of C.
Proof. Let (f1, f2, . . . , fk) be the equations generating the (saturated) ideal our affine chart) identifyφC as
{(α, . . . , ε, a, . . . , h)|f1(P) =· · ·=fk(P) = 0,
a=β−αh, b=γ−αf, c=δ−αg, d=ε−αh}.
Bracing ourselves, we write the Jacobian matrix of this system:
which clearly has corank equal to the corank of the Jacobian matrix of the curve itself atP. HenceφC is singular at and only at points (v, `), wherev∈C is singular.
Consider the projection p2 into G(2,6), onto α(C). Since degC ≥2, this map is everywhere finite. Since the generic line intersecting C will do so in only one point, the degree is 1. Hence this map is birational. Now, by [KUT17, lemma 3.2],α(C) is smooth aty if and only iff−1(y) is a single pointx, and dxf:Tx(X)→Ty(Y) is an injection. Soα(C) is singular at the secant lines of C.
Suppose, then, thatf−1(y) ={x}, and thatdxf is not injective. Sincedxfin our case is projection onto the last 8 coordinates, a nonzero element in the kernel must take the formu=
∗ ∗ ∗ ∗ ∗ 0 . . . 0T. In fact, we see by mul-tiplying with Equation (4.2) that we haveu=
λ eλ f λ gλ hλ . . . 0T