5.2 Lines on hypersurfaces
Next, we prove a well-known result concerning lines on hypersurfaces. The theorem appears in [CG72] for cubics, where it is proved through polynomial substitutions. We give a somewhat more modern proof.
Proposition 5.2.1. LetX ∈Pr+1 be a smooth hypersurface of degree d. Let p(n) be the number of partitions ofn. Then, forr≥dthe lines in X may be classified in p(d−1)classes by their normal bundle. The normal bundle can be written on the form the fact that any vector bundle on a line splits as a direct sum of line bundles,)
5.2. Lines on hypersurfaces We prove the two conditions on the normal sheaf through the exact sequence of normal bundles:
0→Nl/X →Nl/Pn→NX/Pn|l→0.
Indeed, sincel is the zero locus of r linear equations, we have that Nl/Pn = Ol(1)r, and similarlyNX/Pn =OX(d), soNX/Pn|l=Ol(d). Hence the sequence
Now the first claim follows by twisting the sequence by−2 and taking the long exact cohomology sequence. The second claim follows from taking the first Chern classes:
Related to the classification is a simple observation with surprising corollaries.
Proposition 5.2.2. For l a line on an n-dimensional hypersurface X, the infinitesimal deformations of l in X fixing a point has dimension equal to h0(Nl/X(−1)). the Hom-scheme parametrizes every map fromP1intoX, whereas we are only interested in its image. So applying any automorphism ofP1fixingpto the line will give the same image inX. But the automorphism group of a projective line isPGL(2), and the subgroup of this fixing a point has dimension 2, and we are done. Hence the tangent space to [l] in the scheme of lines passing through a pointpshould have dimensionh0(Nl/X(−1)).
Corollary 5.2.3. For a general hypersurfaceX of degreed, of sufficiently great dimensionn, the possible classes number(d+ 1)/2 fordodd,d/2 if dis even.
The general line on such a hypersurface will have normal bundle Od−1⊕ O(1)n. 35
5. Lines on hypersurfaces
Proof. We know by Proposition 2.1.5 that for generalX, the varietyF(X) of lines onX is smooth and has dimension 2n−d−1. Letl⊂X be a line. Then TF(X),l =H0(l,Nl/X). By the classification above, we have
H0(l,Nl/X) = 2n−d−1 only in the cases
Nl/X =Ol(−1)i⊕ Od−1−2il ⊕ Ol(1)n−d+i for alli, of which there are the stated number of classes.
For the second claim, we note that the general point on X should have a n−d-dimensional family of lines through itself. Hence we should have H0(Nl/X(−1)) =n−d, which only happens ifNl/X=Od−1l ⊕ Ol(1)n−d. We will later on give a characterization of the loci of lines with uncommon normal bundle on cubic hypersurfaces.
Let us also here show that:
Lemma 5.2.4. Let P ⊂X ⊂Pn+1 be a k-plane contained in a smooth hyper-surface of dimension n. Then a general hyperplane section ofX containingP will also be smooth as long as n >2k.
Proof. Bézout’s theorem applied to the linear system of hyperplanes passing through P guarantees that the general section is smooth outside the base locus P. To examine what happens alongP, assumeP =Z(xk+1, . . . , xn+1), and let the hyperplane have equationZ(ak+1zk+1+. . . an+1zn+1). Then, lettingF be the equation ofX, it is clear that
F =
n+1
X
i=k+1
xk+1fk+1+ (terms with higher powers ofxk+1, . . . , xn+1) where thefi are forms inx0, . . . xk. AlongP, the Jacobian matrix becomes
0 · · · 0 ak+1 · · · an+1
0 · · · 0 fk+1 · · · fn+1
,
where the firstk+ 1 columns are zero. Since thefi depend onk+ 1 variables, the matrix will have rank 2 at every point inP ifn+ 1−(k+ 1)> k+ 1 - i.e.
n >2k.
It is worthwhile to point out another way of showing the classification of lines by their normal bundles: Let S ∈ P3 be a smooth surface of degree d containing a linel. (Such surfaces do exist, for instance the Fermat surfaces will contain lines.) Then the adjunction formula gives
(Nl/S⊗ωS)|l=ωl, and sinceωl=Ol(−2), ωS=OS(d−4), so
Nl/S =Ol(2−d).
5.2. Lines on hypersurfaces Then we can letX be a degreed-hypersurface of dimensionn, and use induction onn. Let H be a hyperplane, then Lemma 5.2.4 and the short exact sequence
0→Nl/X∩H →Nl/X→N(X∩H)/X|l=Ol(1)→0
will give the same result. This method has the advantage of showing that if a hypersurface contains a line with highly abnormal normal bundle, then an intersection with a general smooth hyperplane containing the line will restrict the possible normal bundles of the line in the intersection.
Indeed, this shows that:
Lemma 5.2.5. In the above context, we have fori= 0,1 that if h1(l,Nl/X(−i)) =k, thenh1(l,Nl/X∩H(−i))≥k.
If we think ofH1(l,Nl/X) as the space of obstructions to lifting abstract deformations ofl to deformations ofl insideX, this fits with our geometric understanding – if it is already difficult to deform a line inside a hypersurface, it should not be easier to deform inside a hyperplane section.
The original result of Clemens and Griffiths was that
Corollary 5.2.6. A line l on a smooth cubic hypersurface has normal sheaf either
• Ol2⊕ Ol(1)n and we say thatl is of type 1, or
• Ol(−1)⊕ Ol(1)n+1 and we say thatl is of type 2.
which follows from Proposition 5.2.1.
This leads to a result which we have used implicitly:
Corollary 5.2.7. The Fano scheme of a cubic hypersurface is smooth.
Proof. Both types of normal sheaves have h0(l,Nl/X) = 2n+ 2. Since this
equals dimT[l](F(X)), we are done.
The analysis of lines and their normal bundles can be carried further than what we have done here.In fact, we have the following result:
Proposition 5.2.8. Let dbe a positive integer and F =L
Ol(ei)a sheaf on P1 fulfilling the conditions of Proposition 5.2.1. There is a hypersurfaceX of degreedcontaining linel such that Nl/X =F.
Proof. See [3264, Prop 6.30]
Corollary 5.2.9. Let X be a cubic hypersurface. Let p∈ X be a point. If there is a 1-dimensional family of lines passing through p, the curve α(p) is singular precisely at all lines of type 2.
Proof. Lines of type 2 have a greater tangent space than normal inα(p).
The classification of lines by their normal bundles also allows us to determine what linear spaces are tangent to hypersurfaces along lines.
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5. Lines on hypersurfaces
Corollary 5.2.10. Let l ⊂ X = Z(f) ⊂ Pn be a line in a hypersurface of degree d. Assume that Nl/X hask O(1)-summands. Then there is a unique k+ 1-plane tangent to X along l. rational curve inPn∨. This curve spans a linear subspace ofPn∨, namely the projectivisation of the subspace V of H0(Ol(d−1)) spanned by the partial derivatives. We interpretPV as the linear system of hyperplanes tangent toX at any point alongl, and so we find the dimension of its base locus. Consider the sequence
0→Nl/X →Nl/Pn →NX/Pn|l→0 the last map of which by Lemma 5.1.1 is given by
(λ2, . . . , λn)7→X
a k+ 1-dimensional base locus.
Especially, we have Clemens and Griffiths’ original application:
Corollary 5.2.11. LetX be a cubic hypersurface of dimensionn. For a line l of type 1 onX, there is a uniquen−2-plane tangent toX alongl If l is of type 2, there is a uniquen−1 plane tangent toX along l.
Recall that we termed a point on a cubic hypersurface an Eckardt point if the family of lines through the point had dimension greater than normal.
Example 5.2.12. Consider the Fermat cubic fourfold. All lines passing through the Eckardt points are of type 2.
To see this, pick such a line l∈F(X), and pick projective coordinates such thatl is the line passing through (1 :−1 : 0 : 0 : 0 : 0) and (0 : 0 :a:b:c:d).
Thenf(l) consists of points on the form
(s2:s2:t2a2:t2b2:t2c2:t2d2) , which clearly is twice a line.
We will see that it is in fact true in general that lines through Eckardt points are type 2.
Several interesting facts about the geometry of cubic fourfolds can be deduced from the classification of lines. As an example, take