Chapter 5

Let \(X\) be a surface in \(\mathbf{P}^{n}, n \geqslant 3,\) defined as the complete intersection of hypersurfaces of degrees \(d_{1}, \ldots, d_{n-2},\) with each \(d_{t} \geqslant 2 .\) Show that for all but finitely many choices of \(\left(n, d_{1}, \ldots, d_{n-2}\right),\) the surface \(X\) is of general type. List the exceptional cases, and where they fit into the classification picture.

Let \(f\) be a rational function on the surface \(X\). Show that it is possible to "resolve the singularities of \(f^{\prime \prime}\) in the following sense: there is a birational morphism \(g:\) \(X^{\prime} \rightarrow X\) so that \(f\) induces a morphism of \(X^{\prime}\) to \(\mathbf{P}^{1}\). [Hints: Write the divisor of \(f\) as \((f)=\sum n_{i} C_{i} .\) Then apply embedded resolution (3.9) to the curve \(Y=\bigcup C_{i}\) Then blow up further as necessary whenever a curve of zeros meets a curve of poles until the zeros and poles of \(f\) are disjoint.]

Let \(C, D\) be any two divisors on a surface \(X\), and let the corresponding invertible sheaves be \(\mathscr{L}, \mathscr{M} .\) Show that $$C . D=\chi\left(\mathcal{O}_{X}\right)-\chi\left(\mathscr{L}^{-1}\right)-\chi\left(\mathscr{M}^{-1}\right)+\chi\left(\mathscr{L}^{-1} \otimes \mathscr{M}^{-1}\right)$$

Let \(X\) be a nonsingular projective variety of any dimension, let \(Y\) be a nonsingular subvariety, and let \(\pi: \widetilde{X} \rightarrow X\) be obtained by blowing up \(Y\). Show that \(p_{a}(\tilde{X})=\) \(p_{a}(X)\)

If \(X\) is a birationally ruled surface, show that the curve \(C\), such that \(X\) is birationally equivalent to \(C \times \mathbf{P}^{1}\), is unique (up to isomorphism)

Prove the following theorem of Chern and Griffiths. Let \(X\) be a nonsingular surface of degree \(d\) in \(P_{c}^{n+1},\) which is not contained in any hyperplane. If \(d<2 n\), then \(p_{g}(X)=0 .\) If \(d=2 n,\) then either \(p_{\theta}(X)=0,\) or \(p_{g}(X)=1\) and \(X\) is a \(\mathrm{K} 3\) surface. \([\text {Hint}: \text { Cut } X\) with a hyperplane and use Clifford's theorem (IV, 5.4). For the last statement, use the Riemann-Roch theorem on \(X\) and the Kodaira vanishing theorem (III, 7.15).]

Let \(H\) be a very ample divisor on the surface \(X,\) corresponding to a projective embedding \(X \subseteq \mathbf{P}^{N} .\) If we write the Hilbert polynomial of \(X\) (III, Ex. 5.2) as \\[ F(z)=\frac{1}{2} a z^{2}+b z+c \\] show that \(a=H^{2}, b=\frac{1}{2} H^{2}+1-\pi,\) where \(\pi\) is the genus of a nonsingular curve representing \(H,\) and \(c=1+p_{a} .\) Thus the degree of \(X\) in \(\mathbf{P}^{N},\) as defined in \((\mathrm{I}, \S 7)\) is just \(H^{2} .\) Show also that if \(C\) is any curve in \(X\), then the degree of \(C\) in \(\mathbf{P}^{N}\) is just \(C . H\)

Let \(Y \cong \mathbf{P}^{1}\) be a curve in a surface \(X,\) with \(Y^{2}<0 .\) Show that \(Y\) is contractible \((5.7 .2)\) to a point on a projective variety \(X_{0}\) (in general singular).

Let \(C\) and \(D\) be curves on a surface \(X,\) meeting at a point \(P .\) Let \(\pi: \tilde{X} \rightarrow X\) be the monoidal transformation with center \(P .\) Show that \(\dot{C} . \tilde{D}=C . D-\mu_{P}(C) \cdot \mu_{P}(D)\) Conclude that \(C . D=\sum \mu_{P}(C) \cdot \mu_{P}(D),\) where the sum is taken over all intersection points of \(C\) and \(D\), including infinitely near intersection points.

Recall that the arithmetic genus of a projective scheme \(D\) of dimension 1 is defined as \(p_{a}=1-\chi\left(\mathcal{O}_{D}\right)(\mathrm{III}, \mathrm{Ex} .5 .3)\) (a) If \(D\) is an effective divisor on the surface \(X\), use (1.6) to show that \(2 p_{a}-2=\) \(D \cdot(D+K)\) (b) \(p_{a}(D)\) depends only on the linear equivalence class of \(D\) on \(X\) (c) More generally, for any divisor \(D\) on \(X\), we define the virtual arithmetic genus (which is equal to the ordinary arithmetic genus if \(D\) is effective) by the same formula: \(2 p_{a}-2=D \cdot(D+K) .\) Show that for any two divisors \(C, D\) we have \\[ p_{a}(-D)=D^{2}-p_{a}(D)+2 \\] and \\[ p_{a}(C+D)=p_{a}(C)+p_{a}(D)+C . D-1 \\]

If \(\pi: \tilde{X} \rightarrow X\) is a monoidal transformation with center \(P,\) show that \(H^{1}\left(\tilde{X}, \Omega_{\mathcal{X}}\right) \cong\) \(H^{1}\left(X, \Omega_{X}\right) \oplus k .\) This gives another proof of \((5.8) .[\) Hints: Use the projection formula (III, Ex.8.3) and (III, Ex.8.1) to show that \(H^{\prime}\left(X, \Omega_{x}\right) \cong H^{i}\left(\tilde{X}, \pi^{*} \Omega_{x}\right)\) for each i. Next use the exact sequence \\[ 0 \rightarrow \pi^{*} \Omega_{x} \rightarrow \Omega_{\hat{\mathcal{R}}} \rightarrow \Omega_{\hat{x} / x} \rightarrow 0 \\] and a local calculation with coordinates to show that there is a natural isomorphism \(\Omega_{\hat{X} / X} \cong \Omega_{E},\) where \(E\) is the exceptional curve. Now use the cohomology sequence of the above sequence (you will need every term) and Serre duality to get the result.]

Let \(\pi: \tilde{X} \rightarrow X\) be a monoidal transformation, and let \(D\) be a very ample divisor on \(X .\) Show that \(2 \pi^{*} D-E\) is ample on \(\tilde{X} .[\) Hint: Use a suitable generalization of \(\left.(\mathrm{I}, \mathrm{Ex} .7 .5) \text { to curves in } \mathbf{P}^{n} .\right]\)

(a) If \(\delta\) is a locally free sheaf of rank \(r\) on a (nonsingular) curve \(C\), then there is a sequence $$0=\mathscr{E}_{0} \subseteq \mathscr{E}_{1} \subseteq \ldots \subseteq \mathscr{E}_{r}=\mathscr{E}$$ of subsheaves such that \(\mathscr{E}_{i} / \mathscr{E}_{i-1}\) is an invertible sheaf for each \(i=1, \ldots, \mathrm{r}\). We say that \(\mathscr{E}\) is a successive extension of invertible sheaves. [Hint: Use(II, Ex. 8.2 ).] (b) Show that this is false for varieties of dimension \(\geqslant 2 .\) In particular, the sheaf of differentials \(\Omega\) on \(\mathbf{P}^{2}\) is not an extension of invertible sheaves.

(a) If a surface \(X\) of degree \(d\) in \(\mathbf{P}^{3}\) contains a straight line \(C=\mathbf{P}^{1},\) show that \(C^{2}=2-d\) (b) Assume char \(k=0,\) and show for every \(d \geqslant 1,\) there exists a nonsingular surface \(X\) of degree \(d\) in \(\mathbf{P}^{3}\) containing the line \(x=y=0\)

Multiplicity of a Local Ring. (See Nagata \([7, \mathrm{Ch} \text { III, } \S 23]\) or Zariski-Samuel \([1, \text { vol } 2, \mathrm{Ch} \text { VIII, } \S 10] .\) Let \(A\) be a noetherian local ring with maximal ideal \(m.\) For any \(l>0,\) let \(\psi(l)=\operatorname{length}\left(A / m^{2}\right) .\) We call \(\psi\) the Hilbert -Samuel function of \(A\). (a) Show that there is a polynomial \(P_{A}(z) \in \mathbf{Q}[z]\) such that \(P_{A}(l)=\psi(l)\) for all \(l \gg 0 .\) This is the Hilbert-Samuel polynomial of \(A .\) [Hint: Consider the graded \(\left.\operatorname{ring} \operatorname{gr}_{m} A=\oplus_{d \geqslant 0} m^{d} / m^{d+1}, \text { and apply }(1,7.5) .\right]\) (b) Show that \(\operatorname{deg} P_{A}=\operatorname{dim} A\) (c) Let \(n=\operatorname{dim} A .\) Then we define the multiplicity of \(A\), denoted \(\mu(A),\) to be \((n !)\) (leading coefficient of \(P_{A}\) ). If \(P\) is a point on a noetherian scheme \(X\), we define the multiplicity of \(P\) on \(X, \mu_{P}(X),\) to be \(\mu\left(\mathcal{O}_{P, X}\right)\) (d) Show that for a point \(P\) on a curve \(C\) on a surface \(X,\) this definition of \(\mu_{P}(C)\) coincides with the one in the text just before \((3.5 .2)\) (e) If \(Y\) is a variety of degree \(d\) in \(\mathbf{P}^{n}\), show that the vertex of the cone over \(Y\) is a point of multiplicity \(d\)

Let \(C\) be a curve of genus \(g\), and let \(X\) be the ruled surface \(C \times \mathbf{P}^{1}\). We consider the question, for what integers \(s \in \mathbf{Z}\) does there exist a section \(D\) of \(X\) with \(D^{2}=>?\) First show that \(s\) is always an even integer, say \(s=2 r\) (a) Show that \(r=0\) and any \(r \geqslant g+1\) are always possible. Cf. (IV, Ex. 6.8 ). (b) If \(g=3,\) show that \(r=1\) is not possible, and just one of the two values \(r=2,3\) is possible, depending on whether \(C\) is hyperelliptic or not.

Let \(C\) be a curve, and let \(\pi: X \rightarrow C\) and \(\pi^{\prime}: X^{\prime} \rightarrow C\) be two geometrically ruled surfaces over \(C .\) Show that there is a finite sequence of elementary transformations (5.7.1) which transform \(X\) into \(X^{\prime}\). [Hints: First show if \(D \subseteq X\) is a section of \(\pi\) containing a point \(P,\) and if \(\tilde{D}\) is the strict transform of \(D\) by \(\mathrm{elm}_{p},\) then \(\tilde{D}^{2}=D^{2}-1\) (Fig. 23 ). Next show that \(X\) can be transformed into a geometrically ruled surface \(X^{\prime \prime}\) with invariant \(e \gg 0 .\) Then use \((2.12),\) and study how the ruled surface \(\mathbf{P}(\mathcal{E})\) with \(\mathscr{E}\) decomposable behaves under \(\mathrm{elm}_{\mathrm{P}} .\)]

Let \(a_{1}, \ldots, a_{r}, r \geqslant 5,\) be distinct elements of \(k,\) and let \(C\) be the curve in \(\mathbf{P}^{2}\) given by the (affine) equation \(y^{2}=\prod_{i=1}^{r}\left(x-a_{i}\right) .\) Show that the point \(P\) at infinity on the \(y\) -axis is a singular point. Compute \(\delta_{P}\) and \(g(\tilde{Y}),\) where \(\tilde{Y}\) is the normalization of \(Y\). Show in this way that one obtains hyperelliptic curves of every genus \(g \geqslant 2\)

Values of \(e .\) Let \(C\) be a curve of genus \(g \geqslant 1\) (a) Show that for each \(0 \leqslant e \leqslant 2 g-2\) there is a ruled surface \(X\) over \(C\) with invariant \(e,\) corresponding to an indecomposable \(\mathscr{E}\). Cf. (2.12) (b) Let \(e<0,\) let \(D\) be any divisor of degree \(d=-e,\) and let \(\check{\xi} \in H^{1}(\mathscr{L}(-D))\) be a nonzero element defining an extension $$0 \rightarrow \mathscr{C}_{C} \rightarrow \mathscr{E} \rightarrow \mathscr{L}(D) \rightarrow 0$$ Let \(H \subseteq|D+K|\) be the sublinear system of codimension 1 defined by ker \(\check{\zeta}\) where \(\check{\text { ? is considered as a linear functional on } H^{0}(\mathscr{L}(D+K)) . \text { For any }}\) effective divisor \(E\) of degree \(d-1\), let \(L_{E} \subseteq|D+K|\) be the sublinear system \(|D+K-E|+E .\) Show that \(\mathscr{E}\) is normalized if and only if for each \(E\) as above, \(L_{E} \neq H .\) Cf. proof of (2.15) (c) Now show that if \(-g \leqslant e<0\), there exists a ruled surface \(X\) over \(C\) with invariant \(e .[\text {Hint}: \text { For any given } D \text { in (b), show that a suitable } \xi\) exists, using an argument similar to the proof of (II, 8.18 ). (d) For \(g=2\), show that \(e \geqslant-2\) is also necessary for the existence of \(X\). Note. It has been shown that \(e \geqslant-g\) for any rulèd surface (Nagata [8] ).

(a) If \(C\) is a curve of genus \(g\), show that the diagonal \(\Delta \subseteq C \times C\) has self-intersection \(\Delta^{2}=2-2 g .\) (Use the definition of \(\Omega_{C / k}\) in (II, \(\S 8\) ).) (b) Let \(l=C \times\) pt and \(m=\mathrm{pt} \times C .\) If \(g \geqslant 1,\) show that \(l, m,\) and \(\Delta\) are linearly independent in \(\mathrm{Num}(C \times C) .\) Thus \(\mathrm{Num}(C \times C)\) has rank \(\geqslant 3,\) and in particular, \(\operatorname{Pic}(C \times C) \neq p_{1}^{*}\) Pic \(C \oplus p_{2}^{*}\) Pic \(C .\) Cf. (III, Ex. 12.6 ), (IV, Ex. 4.10 ).

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?

Show that every locally free sheaf of finite rank on \(\mathbf{P}^{1}\) is isomorphic to a direct sum of invertible sheaves. [Hint: Choose a subinvertible sheaf of maximal degree. and use induction on the rank.

Algebraic Equivalence of Divisors. Let \(X\) be a 'surface. Recall that we have defined an algebraic family of effective divisors on \(X,\) parametrized by a nonsingular curve \(T,\) to be an effective Cartier divisor \(D\) on \(X \times T,\) flat over \(T\) (III, 9.8.5). In this case, for any two closed points \(0,1 \in T\), we say the corresponding divisors \(D_{0}, D_{1}\) on \(X\) are prealgebraically equivalent. Two arbitrary divisors are prealgebraically equivalent if they are differences of prealgebraically equivalent effective divisors. Two divisors \(D, D^{\prime}\) are algebraically equivalent if there is a finite sequence \(D=D_{0}, D_{1}, \ldots, D_{n}=D^{\prime}\) with \(D_{i}\) and \(D_{i+1}\) prealgebraically equivalent for each \(i\) (a) Show that the divisors algebraically equivalent to 0 form a subgroup of Div \(X\) (b) Show that linearly equivalent divisors are algebraically equivalent. [Hint: If \((f)\) is a principal divisor on \(X,\) consider the principal divisor \((t f-u)\) on \(X \times \mathbf{P}^{1}\) where \(t, u\) are the homogeneous coordinates on \(\mathbf{P}^{1}\).] (c) Show that algebraically equivalent divisors are numerically equivalent. [Hint: Use (III, 9.9) to show that for any very ample \(H,\) if \(D\) and \(D^{\prime}\) are algebraically equivalent, then \(\left.D . H=D^{\prime} . H .\right]\) Note. The theorem of Néron and Severi states that the group of divisors modulo algebraic equivalence, called the Néron-Severi group, is a finitely generated abelian group. Over \(\mathbf{C}\) this can be proved easily by transcendental methods \((\mathrm{App} . \mathrm{B}, \S 5)\) or as in (Ex. 1.8 ) below. Over a field of arbitrary characteristic, see Lang and Néron [1] for a proof, and Hartshorne [6] for further discussion. since Num \(X\) is a quotient of the Néron-Severi group, it is also finitely generated, and hence free, since it is torsion-free by construction.

Let \(Y\) be an irreducible curve on a surface \(X,\) and suppose there is a morphism \(f: X \rightarrow X_{0}\) to a projective variety \(X_{0}\) of dimension \(2,\) such that \(f(Y)\) is a point \(P\) and \(f^{-1}(P)=Y .\) Then show that \(Y^{2}<0 .[\text { Hint: Let }|H|\) be a very ample (Cartier) divisor class on \(X_{0},\) let \(H_{0} \in|H|\) be a divisor containing \(P,\) and let \(H_{1} \in|H|\) be a divisor not containing \(\left.P . \text { Then consider } f^{*} H_{0}, f^{*} H_{1} \text { and } \hat{H}_{0}=f^{*}\left(H_{0}-P\right)^{-} .\right]\)

For each of the following singularities at (0,0) in the plane, give an embedded resolution, compute \(\delta_{P},\) and decide which ones are equivalent. (a) \(x^{3}+y^{5}=0\) (b) \(x^{3}+x^{4}+y^{5}=0\) (c) \(x^{3}+y^{4}+y^{5}=0\) (d) \(x^{3}+y^{5}+y^{6}=0\) (e) \(x^{3}+x y^{3}+y^{5}=0\)

If \(D\) is any divisor of degree \(d\) on the cubic surface \((4.7 .3),\) show that $$p_{a}(D) \leqslant\left\\{\begin{array}{ll} \frac{1}{6}(d-1)(d-2) & \text { if } d \equiv 1,2(\bmod 3) \\ \frac{1}{6}(d-1)(d-2)+\frac{2}{3} & \text { if } d \equiv 0(\bmod 3). \end{array}\right.$$ Show furthermore that for every \(d > 0\), this maximum is achieved by some irreducible nonsingular curve.

On the elliptic ruled surface \(X\) of \((2.11 .6),\) show that the sections \(C_{0}\) with \(C_{0}^{2}=1\) form a one-dimensional algebraic family, parametrized by the points of the base curve \(C,\) and that no two are linearly equivalent.

Cohomology Class of a Divisor. For any divisor \(D\) on the surface \(X\), we define its cohomology class \(c(D) \in H^{1}\left(X, \Omega_{X}\right)\) by using the isomorphism Pic \(X \cong\) \(H^{1}\left(X, \mathcal{O}_{X}^{*}\right)\) of \((\mathrm{III}, \mathrm{Ex} .4 .5)\) and the sheaf homomorphism \(d \log : \mathscr{C}^{*} \rightarrow \Omega_{X}(\mathrm{III}\) Ex. \(7.4 \mathrm{c}\) ). Thus we obtain a group homomorphism \(c: \operatorname{Pic} X \rightarrow H^{1}\left(X, \Omega_{X}\right) .\) On the other hand, \(H^{1}(X, \Omega)\) is dual to itself by Serre duality (III, 7.13), so we have a nondegenerate bilinear map $$\langle\quad, \quad\rangle: H^{1}(X, \Omega) \times H^{1}(X, \Omega) \rightarrow k$$ (a) Prove that this is compatible with the intersection pairing, in the following sense: for any two divisors \(D, E\) on \(X,\) we have \\[ \langle c(D), c(E)\rangle=(D . E) \cdot 1 \\] in \(k .[\text { Hint }: \text { Reduce to the case where } D \text { and } E\) are nonsingular curves meeting transversally. Then consider the analogous map \(c:\) Pic \(D \rightarrow H^{1}\left(D, \Omega_{D}\right),\) and the fact (III, Ex. 7.4) that \(c\) (point) goes to 1 under the natural isomorphism of \(\left.H^{1}\left(D, \Omega_{D}\right) \text { with } k .\right]\) (b) If char \(k=0,\) use the fact that \(H^{1}\left(X, \Omega_{X}\right)\) is a finite-dimensional vector space to show that \(\mathrm{Num} X\) is a finitely generated free abelian group.

A surface singularity. Let \(k\) be an algebraically closed field, and let \(X\) be the surface in \(\mathbf{A}_{k}^{3}\) defined by the equation \(x^{2}+y^{3}+z^{5}=0 .\) It has an isolated singularity at the origin \(P=(0,0,0)\) (a) Show that the affine ring \(A=k[x, y, z] /\left(x^{2}+y^{3}+z^{5}\right)\) of \(X\) is a unique factorization domain, as follows. Let \(t=z^{-1} ; u=t^{3} x,\) and \(v=t^{2} y .\) Show that \(z\) is irreducible in \(A ; t \in k[u, v],\) and \(A\left[z^{-1}\right]=k\left[u, v, t^{-1}\right] .\) Conclude that \(A\) is a UFD. (b) Show that the singularity at \(P\) can be resolved by eight successive blowings-up. If \(\tilde{X}\) is the resulting nonsingular surface, then the inverse image of \(P\) is a union of eight projective lines, which intersect each other according to the Dynkin \(\operatorname{diagram} \mathbf{E}_{\mathbf{s}}\). Here each circle denotes a line, and two circles are joined by a line segment whenever the corresponding lines intersect. Note. This singularity has interesting connections with local algebra, invariant theory, and topology. In case \(k=\mathbf{C},\) Mumford [6] showed that the completion \(\hat{A}\) of the ring \(A\) at the maximal ideal \(\mathrm{m}=(x, y, z)\) is also a UFD. This is remarkable, because in general the completion of a local UFD need not be UFD, although the converse is true (theorem of Mori) - see Samuel \([3] .\) Brieskorn [2] showed that the corresponding analytic local ring \(\mathbf{C}\\{x, y, z\\} /\left(x^{2}+y^{3}+z^{5}\right)\) is the only nonregular normal 2 -dimensional analytic local ring which is a UFD. Lipman [2] generalized this as follows: over any algebraically closed field \(k\) of characteristic \(\neq 2,3,5,\) the only nonregular normal complete 2 -dimensional local ring which is a UFD is \(k[[x, y, z]] /\left(x^{2}+y^{3}+z^{5}\right)\) See also Lipman [3] for a report on recent work connected with UFD's. This singularity arose classically out of Klein's work on the icosahedron. The group \(I\) of rotations of the icosahedron, which is isomorphic to the simple group of order \(60,\) acts naturally on the 2 -sphere. Identifying the 2 -sphere with \(\mathbf{P}_{\mathrm{C}}^{1}\) by stereographic projection, the group \(I\) appears as a finite subgroup of Aut \(\mathbf{P}_{\mathbf{C}}^{1}\). This action lifts to give an action of the binary icosahedral group \(\bar{I}\) on \(\mathbf{C}^{2}\) by linear transformations of the complex variables \(t_{1}\) and \(t_{2} .\) Klein \([2, \mathrm{I}, 2,813, \text { p.62 }]\) found three invariant polynomials \(x, y, z\) in \(t_{1}\) and \(t_{2},\) related by the equation \(x^{2}+y^{3}+\) \(z^{5}=0 .\) Thus the surface \(X\) appears as the quotient of \(\mathbf{A}_{\mathbf{C}}^{2}\) by the action of the group \(\bar{I}\). In particular, the local fundamental group of \(X\) at \(P\) is just \(\bar{I}\). With regard to the topology of algebraic varieties over \(C,\) Mumford [6] showed that a normal algebraic surface over \(\mathbf{C}\), whose underlying topological space (in its "usual" topology) is a topological manifold, must be nonsingular. Brieskorn showed that this is not so in higher dimensions. For example, the underlying topological space of the hypersurface in \(\mathbf{C}^{4}\) defined by \(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{3}=0\) is a manifold. Later Brieskorn [1] showed that if one intersects such a singularity with a small sphere around the singular point, then one may get a topological sphere whose differentiable structure is not the standard one. Thus for example, by intersecting the singularity \\[ x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{3}+x_{5}^{6 k-1}=0 \\] in \(\mathbf{C}^{5}\) with a small sphere around the origin, for \(k=1,2, \ldots, 28,\) one obtains all 28 possible differentiable structures on the 7 -sphere. See Hirzebruch and Mayer [1] for an account of this work.

Show that the following two singularities have the same multiplicity, and the same configuration of infinitely near singular points with the same multiplicities, hence the same \(\delta_{P},\) but are not equivalent. (a) \(x^{4}-x y^{4}=0\) (b) \(x^{4}-x^{2} y^{3}-x^{2} y^{5}+y^{8}=0\).

Show that a divisor class \(D\) on the cubic surface contains an irreducible curve \(\Leftrightarrow\) it contains an irreducible nonsingular curve \(\Leftrightarrow\) 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 \(D^{2} > 0 .[\text { Hint : Generalize }(4.11)\) 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 \(\left.\mathbf{P}^{1} \times \mathbf{P}^{1} \text { and the rational ruled surface } X_{1},(2.18) .\right]\)

A locally free sheaf \(\mathscr{E}\) on a curve \(C\) is said to be stable if for every quotient locally free sheaf \(\mathscr{E} \rightarrow \mathscr{F} \rightarrow 0, \mathscr{F} \neq \mathscr{E}, \mathscr{F} \neq 0,\) we have \\[ (\operatorname{deg} \mathscr{F}) / \operatorname{rank} \mathscr{F}>(\operatorname{deg} \delta) / \operatorname{rank} \delta \\] Replacing \(>\) by \(\geqslant\) defines semistable (a) A decomposable \(\mathscr{E}\) is never stable (b) If \(\mathscr{E}\) has rank 2 and is normalized, then \(\mathscr{E}\) is stable (respectively, semistable) if and only if \(\operatorname{deg} \mathscr{E}>0\) (respectively, \(\geqslant 0\) ). (c) Show that the indecomposable locally free sheaves \(\mathscr{E}\) of rank 2 that are not semistable are classified, up to isomorphism, by giving (1) an integer 0 \(\leq 0 \leq\) \(2 g-2,(2)\) an element \(\mathscr{L} \in\) Pic \(C\) of degree \(-e,\) and (3) a nonzero \(\xi \in H^{1}\left(\mathscr{H}^{-}\right)\) determined up to a nonzero scalar multiple.

(a) If \(H\) is an ample divisor on the surface \(X\), and if \(D\) is any divisor, show that \\[ \left(D^{2}\right)\left(H^{2}\right) \leqslant(D . H)^{2} \\] (b) Now let \(X\) be a product of two curves \(X=C \times C^{\prime} .\) Let \(l=C \times p t,\) and \(m=\mathrm{pt} \times C^{\prime} .\) For any divisor \(D\) on \(X,\) let \(a=D . l, b=D . m .\) Then we say \(D\) has type \((a, b) .\) If \(D\) has type \((a, b),\) with \(a, b \in \mathbf{Z},\) show that \\[ D^{2} \leqslant 2 a b \\] and equality holds if and only if \(D \equiv b l+a m .\) [Hint: Show that \(H=l+m\) is ample, let \(E=l-m\), let \(D^{\prime}=\left(H^{2}\right)\left(E^{2}\right) D-\left(E^{2}\right)(D . H) H-\left(H^{2}\right)(D . E) E,\) and apply (1.9). This inequality is due to Castelnuovo and Severi. See Grothendieck \([2] .]\)

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

Let \(Y\) be a nonsingular curve on a quadric cone \(X_{0}\) in \(\mathbf{P}^{3}\). Show that either \(Y\) is a complete intersection of \(X_{0}\) with a surface of degree \(a \geqslant 1\), in which case deg \(Y=\) \(2 a, g(Y)=(a-1)^{2},\) or, deg \(Y\) is odd, say \(2 a+1,\) and \(g(Y)=a^{2}-a\) \((\mathrm{IV}, 6.4 .1) .[\text { Hint }: \text { Use }(2.11 .4) .]\)

In this problem, we assume that \(X\) is a surface for which \(\mathrm{Num} X\) is finitely generated (i.e., any surface, if you accept the Néron-Severi theorem (Ex. 1.7 )). (a) If \(H\) is an ample divisor on \(X\), and \(d \in \mathbf{Z}\), show that the set of effective divisors \(D\) with \(D . H=d,\) modulo numerical equivalence, is a finite set. [Hint: Use the adjunction formula, the fact that \(p_{a}\) of an irreducible curve is \(\geqslant 0,\) and the fact that the intersection pairing is negative definite on \(\left.H^{\perp} \text { in } \mathrm{Num} X .\right]\) (b) Now let \(C\) be a curve of genus \(g \geqslant 2\), and use (a) to show that the group of automorphisms of \(C\) is finite, as follows. Given an automorphism \(\sigma\) of \(C\), let \(\Gamma \subseteq X=C \times C\) be its graph. First show that if \(\Gamma \equiv \Delta\), then \(\Gamma=\Delta\), using the fact that \(\Delta^{2}<0,\) since \(g \geqslant 2\) (Ex. 1.6 ). Then use (a). Cf. (IV, Ex. 2.5)

The Weyl Groups. 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 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 \((12),(23), \ldots,(n-1, n)\) 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 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 \(\mathbf{E}_{6} \cong G\). Note: See Manin \([3, \$ 25,26]\) for more about Weyl groups, root systems and exceptional curves

Let \(X\) be a ruled surface over the curve \(C\), defined by a normalized bundle \(\mathscr{E}\) and let e be the divisor on \(C\) for which \(\mathscr{L}(\mathrm{e}) \cong \wedge^{2} \mathscr{E}(2.8 .1) .\) Let \(\mathrm{b}\) be any divisor on \(C\) (a) If \(|b|\) and \(|b+c|\) have no base points, and if \(b\) is nonspecial, then there is a section \(D \sim C_{0}+\mathrm{b} f,\) and \(|D|\) has no base points (b) If \(b\) and \(b+c\) are very ample on \(C,\) and for every point \(P \in C,\) we have \(b-P\) and \(b+e-P\) nonspecial, then \(C_{0}+\) b \(f\) is very ample.

If \(D\) is an ample divisor on the surface \(X,\) and \(D^{\prime} \equiv D,\) then \(D^{\prime}\) is also ample. Give an example to show, however, that if \(D\) is very ample, \(D^{\prime}\) need not be very ample.

Let \(X\) be a ruled surface with invariant \(e\) over an elliptic curve \(C,\) and let b be a divisor on \(C\) (a) If \(\operatorname{deg} b \geqslant e+2,\) then there is a section \(D \sim C_{0}+b f\) such that \(|D|\) has no base points. (b) The linear system \(| C_{0}+\) b \(f |\) is very ample if and only if \(\operatorname{deg} b \geqslant e+3\) Note. The case \(e=-1\) will require special attention.

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 \((4.7 .1))\)

Let \(X\) be a ruled surface over a curve \(C\) of genus \(g,\) with invariant \(e<0,\) and assume that char \(k=p>0\) and \(g \geqslant 2\) (a) If \(Y \equiv a C_{0}+b f\) is an irreducible curve \(\neq C_{0}, f,\) then either \(a=1, b \geqslant 0,\) or \(2 \leqslant a \leqslant p-1, b \geqslant \frac{1}{2} a e,\) or \(a \geqslant p, b \geqslant \frac{1}{2} a e+1-g\) (b) If \(a>0\) and \(b>a\left(\frac{1}{2} e+(1 / p)(g-1)\right),\) then any divisor \(D \equiv a C_{0}+b f\) is ample. On the other hand, if \(D\) is ample, then \(a>0\) and \(b>\frac{1}{2} a e\)

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 \(P_{t}\) (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 \(P_{1}, \ldots, P,\) 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: Prove a lemma that when \(k\) is uncountable, a variety cannot be equal to the union of a countable family of proper closed subsets.] (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 \(C^{2}=-1 .[\text { Hint: Let } L\) 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 \([5, \mathrm{II}, \mathrm{p} .283]\).

Funny behavior in characteristic \(p\). Let \(C\) be the plane curve \(x^{3} y+y^{3} z+z^{3} x=0\) over a field \(k\) of characteristic \(3(\mathrm{IV}, \mathrm{Ex} .2 .4)\) (a) Show that the action of the \(k\) -linear Frobenius morphism \(f\) on \(H^{1}\left(C, \mathcal{O}_{c}\right)\) is identically \(0(\mathrm{Cf} .(\mathrm{IV}, 4.21))\) (b) Fix a point \(P \in C,\) and show that there is a nonzero \(\xi \in H^{1}(\mathscr{L}(-P))\) such that \(f^{*} \xi=0\) in \(H^{1}(\mathscr{L}(-3 P))\) (c) Now let \(\mathscr{E}\) be defined by \(\xi\) as an extension \\[ 0 \rightarrow \mathscr{C}_{\mathrm{c}} \rightarrow \mathscr{E} \rightarrow \mathscr{L}(P) \rightarrow 0 \\] and let \(X\) be the corresponding ruled surface over \(C .\) Show that \(X\) contains a nonsingular curve \(Y \equiv 3 C_{0}-3 f,\) such that \(\pi: Y \rightarrow C\) is purely inseparable. Show that the divisor \(D=2 C_{0}\) satisfies the hypotheses of \((2.21 b),\) but is not ample

Let \(C\) be a nonsingular affine curve. Show that two locally free sheaves \(\mathscr{E}, \mathscr{E}^{\prime}\) of the same rank are isomorphic if and only if their classes in the Grothendieck group \(K(X)(\mathrm{II}, \mathrm{Ex} .6 .10)\) and \((\mathrm{II}, \mathrm{Ex} .6 .11)\) are the same. This is false for a projective curve.