For several reasons, it is easier to investigate the geometry of Eckardt points on cubic hypersurfaces than for higher-degree hypersurfaces. First of all, we obtain other characterizations that do not immediately extend to hypersurfaces of higher degree.
Proposition 6.2.1. LetX ∈Pn+1 be a smooth cubic hypersurface. Recall that sincedimF1(X) = 2n−4, there is through a general point inXan−3-dimension family of lines. Then the Eckardt points onX can also be characterized as:
• flex pointsif n= 1
• points where F1(x)p is a cubic hypersurface of dimensionn−2 if n≥2.
• points where dimF1(X)p=n−2if n≥3.
This makes our choice of terminology acceptable. Classically, a cubic surface is said to have an Eckardt point where three of its lines meet in a point ([Dol12]).1
Remark 6.2.2. We can also see that for a reduced cubic hypersurface inP1(that is, a union of three distinct points), every point is Eckardt - the equation can always be written on the formx20x1−x31.
1Oddly, the given names of prof. Eckardt seem lost to history, except that his initials wereF.E.
6.2. Eckardt points on cubic hypersurfaces Proof of Proposition 6.2.1. Suppose that X is a smooth cubic (i.e. elliptic) curve inP2. First, note that aflex pointon a curve is a point where the tangent line has intersection multiplicity 3 or more [3264, p. 266]. Intuitively, this is where the tangent line to the curve "crosses" the curve.
Now letX be a cubic hypersurface of dimension≥2, withp= (1 : 0 :· · ·: 0)∈X an Eckardt point. The Taylor expansion is then
f =x20x1+x0q(x1, x2, . . . , xr) +c(x1, x2, . . . , xr)
, for q, c quadric and cubic forms. From the discussion at Lemma 4.1.4, we see that F1(X)p is (at least set-theoretically) given by Z(x0)∩Z(q)∩Z(c), which generally has codimension 3. But by definition of Eckardt point, this intersection isZ(x0)∩Z(c)- a cubic hypersurface of one dimension higher.
For the final statement, the same argument as before shows that for an Eckardt point x, F1(X)p is an n−2-dimensional cubic hypersurface. So it remains to be shown that as long as there is ann−2-dimensional family of lines throughp, this family is a cubic hypersurface.
So write again the Taylor expansion off aroundp: f =x20x1+x0g2(x1, . . . , xn) +g3(x1, . . . , xn).
A line inX must be contained inZ(x1, g2, g3). If dimF1(X)p =n−2, then g2(0, x2, . . . , xn), g3(0, x2, . . . , xn) must have common factors. But if the com-mon factor is linear, then X contains a n−1-plane. And if the common factor is quadratic (so equal tog2(0, x2, . . . , xn)), then X contains a quadric n−1-dimensional coneY. But the residual intersection ofX with a hyperplane containingY is also ann−1-plane. However, a smooth hypersurface of dimen-sionn cannot contain any linear space of dimension> n2 (Lemma 5.2.4). So the only conclusion can be that
g2(0, x2, . . . , xn) = 0 sox1divides g2 andpis an Eckardt point.
Harmonic homologies
Eckardt points on a cubic hypersurfaceX can also be characterized through the automorphisms ofX. To state this precisely, the following terminology is useful:
Definition 6.2.3([Dol12, p. 487]). An automorphism of projective space fixing every point in a hyperplane is ahomology. An involutive homology is called a harmonic homology.
Remark 6.2.4. These names come from classical algebraic geometry. Despite the name, there is no clear relation to standard homology theories, and we will only use this terminology in this section.
The following three lemmata are immediate generalizations of those in Dolgachev’s book ([Dol12, section 9.1.4]).
Lemma 6.2.5. An involution ofPn has as fixed locus the union of two linear subspaces of dimensions i, n−i−1.
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6. Eckardt points and lines of type 2
Proof. Letτ be the involution. Interpretingτ as a matrix, we can diagonalise it so that the first i+ 1 elements along the diagonal will be −1, and the other n−i will be 1. Then the fixed subspaces of Pn are Z(x0, . . . , xi) and
Z(xi+1, . . . , xn).
Lemma 6.2.6. Let X ∈ Pn be a cubic hypersurface. There is a bijection between the set of Eckardt points ofX and the set of harmonic homologies of Pn takingX to itself.
Proof. Assume as before that p = (1 : 0 : · · · : 0), so that we can write X =Z(x20x1+x0x1l+c), forl, cforms of degree 1 and 3 inx1, . . . , xn. Under the substitutionx07→x0+2l, the equation ofX becomes
f =x20x1+ec, ec∈C[x1. . . , xn]3. It follows that
ι: (a0:· · ·:an)7→(−a0:· · ·:an)
is an involution with the desired properties. Conversely, assume that κis a harmonic homology takingX to itself, and choose coordinates such thatZ(x0) is the fixed hyperplane. Then (1 : 0· · ·: 0) is the isolated fixed point, and we
may writeX=Z(x20l+c).
This leads to the result:
Lemma 6.2.7. No line in a cubic hypersurface can contain more than two Eckardt points.
Proof. We work by induction, starting with the case whereX is a cubic surface.
Suppose that l ⊂ X contains the Eckardt point p. Let τp be the harmonic homology determined byp, and letH be the fixed hyperplane. Thenllies in the tangent hyperplane ofX atp. Suppose then thatl contains two more Eckardt pointsq, r. ThenH∩X will intersect the tangent planes ofqandrthree times each. It follows thatqandr are both contained inH, a contradiction.
Suppose then thatX is a cubic hypersurface inPn, and assume the lemma for hypersurfaces inPn−1, wheren≥4. Let l⊂X be a line. Following [CC09], we argue as follows: Consider the linear system
L={X∩H|l⊂H,H is a hyperplane}.
This is an−2-dimensional system with fixed locus only l. It follows from Bertini’s theorem ([Har77]) that for a generalP ∈L, the singularities ofP∩X are alongl. But if P∩X is singular atp∈l, thenP =TpX. So this can only happen for a one-dimensional family of P’s and the general member of Lis smooth. Looking a bit ahead, we will know from Lemma 6.6.4 that any Eckardt point onl is an Eckardt point onH∩X. The lemma follows.
How rare are Eckardt points among hypersurfaces of dimensionnand degree d? The answer is, perhaps surprisingly, not dependent on the degree of the hypersurface.
Let X be a degree d hypersurface in Pn. Having an Eckardt point is a codimension n+12
−1-condition on the parameter space of hypersurfaces.
6.3. The variety of lines of type 2
Observation 6.2.8. In all, the family of hypersurfaces of degreedand dimen-sionnwith an Eckardt point has dimension
n+d
Remark 6.2.9. It is clear from the construction that the existence of an Eckardt point is equivalent to saying that X contains a cone over a (smooth) cubic surface. The vertex of this cone is then the Eckardt point. This is yet another characterization.
6.3 The variety of lines of type 2
In this section, we develop the connection between the lines of type 2 on a cubic hypersurface and its Eckardt points. Recall that a linel is of type 2 in the cubicX ifNl/X=Ol(1)2⊕ Ol(−1)n−2. We will see that through an Eckardt point, every line is of type 2. We start by describing the subvarietyS2⊂F(X) representing lines inX of type 2.
Here again,X is a cubic hypersurface inPn+1, Σ the incidence correspon-dence (see Equation (4.1)) with projectionsq: Σ→X, p: Σ→F1(X).
Recall from the proof of Corollary 5.2.10 that the dual map or Gauss map D = DX maps any line l of type 2 onto a nonsingular quadric curve, so degD|l= 2.
Lemma 6.3.1 ([CG72]). S2 has dimension ≤n−2. For a generalX, this dimension is exactlyn−2.
Proof. LetR⊂Σ be the quasiprojective scheme
{([l], x)| |[l]∈S2, x∈l, xnot a ramification point ofDl}.
then we claim: q|R:R→X is finite.
To see this, assume that there is an irreducible curve C ⊂ R such that q(C) =y, wherey∈X is a closed point. The dual mapping on any linel∈S2
is 2 : 1, and so induces an involution on l. Lettingil be this involution, it follows that there is an involutioniR: R→R defined byiR([l], x) = ([l], il(x)).
It follows thatp◦iR =p, and also thatDX◦q◦i=DX◦q. Then, since DX is finite, there must be another pointz∈X (here we use that ([l], x)∈R 45