5_4x

V.4: The Cubic Surface in \({\mathbf{P}}^3\)

V.4.1. #to_work

The linear system of conics in \(\mathbf{P}^2\) with two assigned base points \(P_1\) and \(P_2\) (4.1) determines a morphism \(\psi\) of \(X^{\prime}\) (which is \(\mathbf{P}^2\) with \(P_1\) and \(P_2\) blown up) to a nonsingular quadric surface \(Y\) in \(\mathbf{P}^3\), and furthermore \(X^{\prime}\) via \(\psi\) is isomorphic to \(Y\) with one point blown up.

V.4.2. #to_work

Let \(\varphi\) be the quadratic transformation of \((4.2 .3)\), centered at \(P_1, P_2, P_3\). If \(C\) is an irreducible curve of degree \(d\) in \(\mathbf{P}^2\), with points of multiplicity \(r_1, r_2, r_3\) at \(P_1, P_2, P_3\), then the strict transform \(C^{\prime}\) of \(C\) by \(\varphi\) has degree \begin{align*} d^{\prime}=2 d-r_1-r_2-r_3 \end{align*} and has points of multiplicity

• \(d-r_2-r_3\) at \(Q_1\),
• \(d-r_1-r_3\) at \(Q_2\) and
• \(d-\) \(r_1-r_2\) at \(Q_3\).

The curve \(C\) may have arbitrary singularities. ¹

V.4.3. #to_work

Let \(C\) be an irreducible curve in \(\mathbf{P}^2\). Then there exists a finite sequence of quadratic transformations, centered at suitable triples of points, so that the strict transform \(C^{\prime}\) of \(C\) has only ordinary singularities, i.e., multiple points with all distinct tangent directions (I, Ex. 5.14). Use (3.8).

V.4.4. #to_work

• Use (4.5) to prove the following lemma on cubics: If \(C\) is an irreducible plane cubic curve, if \(L\) is a line meeting \(C\) in points \(P, Q, R\), and \(L^{\prime}\) is a line meeting \(C\) in points \(P^{\prime}, Q^{\prime}, R^{\prime}\), let \(P^{\prime \prime}\) be the third intersection of the line \(P P^{\prime}\) with \(C\), and define \(Q^{\prime \prime}, R^{\prime \prime}\) similarly. Then \(P^{\prime \prime}, Q^{\prime \prime}, R^{\prime \prime}\) are collinear.

• Let \(P_0\) be an inflection point of \(C\), and define the group operation on the set of regular points of \(C\) by the geometric recipe "let the line \(P Q\) meet \(C\) at \(R\), and let \(P_0 R\) meet \(C\) at \(T\), then \(P+Q=T^{\prime \prime}\) as in (II, 6.10.2) and (II, 6.11.4). Use (a) to show that this operation is associative.

V.4.5. #to_work

Prove Pascal’s theorem: if \(A, B, C, A^{\prime}, B^{\prime}, C^{\prime}\) are any six points on a conic, then the points \(P=A B^{\prime} \cdot A^{\prime} B, Q=A C^{\prime} \cdot A^{\prime} C\), and \(R=B C^{\prime} \cdot B^{\prime} C\) are collinear (Fig. 22).

V.4.6. #to_work

Generalize (4.5) as follows: given 13 points \(P_1, \ldots, P_{13}\) in the plane, there are three additional determined points \(P_{14}, P_{15}, P_{16}\), such that all quartic curves through \(P_1, \ldots, P_{13}\) also pass through \(P_{14}, P_{15}, P_{16}\). What hypotheses are necessary on \(P_1, \ldots, P_{13}\) for this to be true?

V.4.7. #to_work

If \(D\) is any divisor of degree \(d\) on the cubic surface (4.7.3), show that \begin{align*} p_a(D) \leqslant \begin{cases}\frac{1}{6}(d-1)(d-2) & \text { if } d \equiv 1,2\, (\bmod 3) \\ \frac{1}{6}(d-1)(d-2)+\frac{2}{3} & \text { if } d \equiv 0\, (\bmod 3)\end{cases} \end{align*} Show furthermore that for every \(d>0\), this maximum is achieved by some irreducible nonsingular curve.

V.4.8. * #to_work

Show that a divisor class \(D\) on the cubic surface contains an irreducible curve \(\iff\) if it contains an irreducible nonsingular curve \(\iff\) it is either

• one of the 27 lines, or

• a conic (meaning a curve of degree 2) with \(D^2=0\), or

• \(D . L \geqslant 0\) for every line \(L\), and \(D^2>0\). ²

V.4.9. #to_work

If \(C\) is an irreducible non-singular curve of degree \(d\) on the cubic surface, and if the genus \(g>0\), then \begin{align*} g \geqslant \begin{cases}\frac{1}{2}(d-6) & \text { if } d \text { is even, } d \geqslant 8 \\ \frac{1}{2}(d-5) & \text { if } d \text { is odd }, d \geqslant 13\end{cases} \end{align*} and this minimum value of \(g>0\) is achieved for each \(d\) in the range given.

V.4.10. #to_work

A curious consequence of the implication (iv) \(\Rightarrow\) (iii) of (4.11) is the following numerical fact: Given integers \(a, b_1, \ldots, b_6\) such that \(b_i>0\) for each \(i, a-b_i-\) \(b_j>0\) for each \(i, j\) and \(2 a-\sum_{i \neq j} b_i>0\) for each \(j\), we must necessarily have \(a^2-\sum b_i^2>0\). Prove this directly (for \(a, b_1, \ldots, b_6 \in \mathbf{R}\) ) using methods of freshman calculus.

V.4.11. The Weyl Groups. #to_work

Given any diagram consisting of points and line segments joining some of them, we define an abstract group, given by generators and relations, as follows:

• Each point represents a generator \(x_i\). The relations are
• \(x_i^2=1\) for each \(i\);
• \(\left(x_i x_j\right)^2=1\) if \(i\) and \(j\) are not joined by a line segment, and
• \(\left(x_i x_j\right)^3=1\) if \(i\) and \(j\) are joined by a line segment.

• The Weyl group \(\mathbf{A}_n\) is defined using the following diagram of \(n-1\) points, each joined to the next: Show that it is isomorphic to the symmetric group \(\Sigma_n\) as follows:

  • Map the generators of \(\mathbf{A}_n\) to the elements \((12),(23), .., (n-1,n)\) of \(\Sigma_n\), to get a surjective homomorphism \(\mathbf{A}_n \rightarrow \Sigma_n\).
  • Then estimate the number of elements of \(\mathbf{A}_n\) to show in fact it is an isomorphism.

• The Weyl group \(\mathbf{E}_6\) is defined using the diagram Call the generators \(x_1, \ldots, x_5\) and \(y\). Show that one obtains a surjective homomorphism \(\mathbf{E}_6 \rightarrow G\), the group of automorphisms of the configuration of 27 lines \((4.10 .1)\), by sending \(x_1, \ldots, x_5\) to the permutations \((12),(23), \ldots,(56)\) of the \(E_i\), respectively, and \(y\) to the element associated with the quadratic transformation based at \(P_1, P_2, P_3\).

• * Estimate the number of elements in \(\mathbf{E}_6\), and thus conclude that \(\mathbf{E}_6 \cong G\). ³

V.4.12. #to_work

Use (4.11) to show that if \(D\) is any ample divisor on the cubic surface \(X\), then \(H^1\left(X, \mathcal{O}_X(-D)\right)=0\). This is Kodaira’s vanishing theorem for the cubic surface (III, 7.15).

V.4.13. #to_work

Let \(X\) be the Del Pezzo surface of degree 4 in \(\mathbf{P}^4\) obtained by blowing up 5.points of \(\mathbf{P}^2(4.7)\)

• Show that \(X\) contains 16 lines.

• Show that \(X\) is a complete intersection of two quadric hypersurfaces in \(\mathbf{P}^4\) (the converse follows from (4.7.1)).

V.4.14. #to_work

Using the method of (4.13.1), verify that there are nonsingular curves in \(\mathbf{P}^3\) with \(d=8, g=6,7 ; d=9, g=7,8,9 ; d=10, g=8,9,10,11\). Combining with (IV, §6), this completes the determination of all posible \(g\) for curves of degree \(d \leqslant 10\) in \(\mathbf{P}^3\).

V.4.15. #to_work

Let \(P_1, \ldots, P_r\) be a finite set of (ordinary) points of \(\mathbf{P}^2\), no 3 collinear. We define an admissible transformation to be a quadratic transformation (4.2.3) centered at some three of the \(P_i\) (call them \(P_1, P_2, P_3\) ).

This gives a new \(\mathbf{P}^2\), and a new set of \(r\) points, namely \(Q_1, Q_2, Q_3\), and the images of \(P_4, \ldots, P_r\).

We say that \(P_1, \ldots, P_r\) are in general position if no three are collinear, and furthermore after any finite sequence of admissible transformations, the new set of \(r\) points also has no three collinear.

• A set of 6 points is in general position if and only if no three are collinear and not all six lie on a conic.

• If \(P_1, \ldots, P_r\) are in general position, then the \(r\) points obtained by any finite sequence of admissible transformations are also in general position.

• Assume the ground field \(k\) is uncountable. Then given \(P_1, \ldots, P_r\) in general position, there is a dense subset \(V \subseteq \mathbf{P}^2\) such that for any \(P_{r+1} \in V, P_1, \ldots, P_{r+1}\) will be in general position. ⁴

• Now take \(P_1, \ldots, P_r \in \mathbf{P}^2\) in general position, and let \(X\) be the surface obtained by blowing up \(P_1, \ldots, P_r\). If \(r=7\), show that \(X\) has exactly 56 irreducible nonsingular curves \(C\) with \(g=0, C^2=-1\), and that these are the only irreducible curves with negative self-intersection. Ditto for \(r=8\), the number being 240 .

• * For \(r=9\), show that the surface \(X\) defined in (d) has infinitely many irreducible nonsingular curves \(C\) with \(g=0\) and \(C^2=-1\). ⁵

V.4.16. #to_work

For the Fermat cubic surface \(x_0^3+x_1^3+x_2^3+x_3^3=0\), find the equations of the 27 lines explicitly, and verify their incidence relations. What is the group of automorphisms of this surface?