## V.4:  The Cubic Surface in $\PP^3$

### 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 
\[
d^{\prime}=2 d-r_1-r_2-r_3
\]
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.[^hint.v.4.2]

[^hint.v.4.2]: 
Hint: Use (Ex. 3.2).

### 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


a. 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.

b. 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).

![](Hartshorne_Problems/5_Hartshorne/figures/2022-10-22_21-13-10.png)


### 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
\[
p_a(D) \leqslant \begin{cases}\frac{1}{6}(d-1)(d-2) & \text { if } d \equiv 1,2\, (\bmod 3) \\ \frac{1}{6}(d-1)(d-2)+\frac{2}{3} & \text { if } d \equiv 0\, (\bmod 3)\end{cases}
\]
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 

a. one of the 27 lines, or 

b. a conic (meaning a curve of degree 2) with $D^2=0$, or 
c. $D . L \geqslant 0$ for every line $L$, and $D^2>0$.[^hint.v.4.8]

[^hint.v.4.8]: 
Hint: Generalize (4.11) to the surfaces obtained by blowing up $2,3,4$, or 5 points of $\mathbf{P}^2$, and combine with our earlier results about curves on $\mathbf{P}^1 \times \mathbf{P}^1$ and the rational ruled surface $X_1,(2.18)$.


### 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
\[
g \geqslant \begin{cases}\frac{1}{2}(d-6) & \text { if } d \text { is even, } d \geqslant 8 \\ \frac{1}{2}(d-5) & \text { if } d \text { is odd }, d \geqslant 13\end{cases}
\]
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.

a. The Weyl group $\mathbf{A}_n$ is defined using the following diagram of $n-1$ points, each joined to the next:
![](Hartshorne_Problems/5_Hartshorne/figures/2022-10-22_21-16-28.png)
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.

b. The Weyl group $\mathbf{E}_6$ is defined using the diagram
![](Hartshorne_Problems/5_Hartshorne/figures/2022-10-22_21-17-05.png)
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 $P_1, P_2, P_3$.

c. \* Estimate the number of elements in $\mathbf{E}_6$, and thus conclude that $\mathbf{E}_6 \cong G$.[^manin_note_v.4.qqq]

[^manin_note_v.4.qqq]: 
Note: See Manin $[3, \S 25,26]$ for more about Weyl groups, root systems, and exceptional curves.


### 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 $\mathbf{P}^2(4.7)$

a. Show that $X$ contains 16 lines.

b. 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. 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.

b. 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.

c. 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.[^hint.v.4.15]

d. 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 .

e. \* For $r=9$, show that the surface $X$ defined in (d) has infinitely many irreducible nonsingular curves $C$ with $g=0$ and $C^2=-1$.[^hint.v.4.15.e]

[^hint.v.4.15.e]: 
Hint: Let $L$ be the line joining $P_1$ and $P_2$. Show that there exist finite sequences of admissible transformations such that the strict transform of $L$ becomes a plane curve of arbitrarily high degree. This example is apparently due to Kodaira-see Nagata $[5, II, p. 283]$.

[^hint.v.4.15]: 
Hint: Prove a lemma that when $k$ is uncountable, a variety cannot be equal to the union of a countable family of proper closed subsets.


### 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?