Show that a divisor class \(D\) on the cubic surface contains an irreducible curve \(\iff\) 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 \(D^2>0\).
Hint: Generalize (4.11) to the surfaces obtained by blowing up \(2,3,4\), or 5 points of \(\PP^2\), and combine with our earlier results about curves on \(\PP^1 \times \PP^1\) and the rational ruled surface \(X_1\), (2.18).
Solution
Let \[ S=X_6=\operatorname{Bl}_{P_1,\ldots,P_6}\PP^2 \] be the cubic surface, with the six points in general position, and let \[ H=-K_S \] be its anticanonical hyperplane class. Thus \[ H^2=3, \] and the \(27\) lines are exactly the \((-1)\)-curves listed in FE-SRFCUBIC.
We first prove the linear-system lemma needed for the sufficiency of (c).
1. For \(2\le r\le6\), write\[ X_r=\operatorname{Bl}_{P_1,\ldots,P_r}\PP^2, \qquad A_r=-K_{X_r}=3h-\sum_{i=1}^r e_i. \]If an integral divisor class \(M\) on \(X_r\) satisfies\[ M\cdot E\ge0 \]for every \((-1)\)-curve \(E\subseteq X_r\), then\[ \boxed{\OO_{X_r}(M)\text{ is globally generated}.} \]
We prove this by induction on \(r\), and for fixed \(r\) by induction on the nonnegative anticanonical degree \[ A_r\cdot M. \] The nonnegativity of this degree follows from the explicit decompositions of \(A_r\) into \((-1)\)-curve classes recorded in step 3 below.
2. Assume \(r>2\) and the assertion knownfor \(X_{r-1}\).
If there is a \((-1)\)-curve \(E\) with \[ M\cdot E=0, \] contract it: \[ \sigma:X_r\longrightarrow X_{r-1}. \] Because \[ \Pic X_r=\sigma^*\Pic X_{r-1}\oplus\ZZ E \] and \(E^2=-1\), the equality \(M\cdot E=0\) says precisely that \[ M=\sigma^*M' \] for a divisor class \(M'\) on \(X_{r-1}\).
Every \((-1)\)-curve \(F'\) on \(X_{r-1}\) avoids the point contracted from \(E\). Indeed, if \(F'\) passed through that point, its strict transform \(F\) would satisfy \[ (-K_{X_r})\cdot F = (-K_{X_{r-1}})\cdot F'-1 =0, \] contradicting ampleness of \(-K_{X_r}\). Hence the strict transform of \(F'\) is a \((-1)\)-curve on \(X_r\), and \[ M'\cdot F'=M\cdot F\ge0. \] The outer induction gives global generation of \(M'\), hence of \(M=\sigma^*M'\).
It remains to treat the case \[ M\cdot E>0 \] for every \((-1)\)-curve \(E\). Since all intersections are integers, \[ M\cdot E\ge1. \] Put \[ N=M-A_r. \] Every \((-1)\)-curve satisfies \[ A_r\cdot E=1, \] so \[ N\cdot E=M\cdot E-1\ge0. \] Step 3 shows that \(A_r\) is a positive integral sum of \((-1)\)-curve classes. Hence \[ A_r\cdot N\ge0. \] Also \[ A_r\cdot N=A_r\cdot M-A_r^2<A_r\cdot M. \] The inner induction therefore makes \(N\) globally generated. Since \(A_r\) is very ample for the del Pezzo surfaces of degree \[ A_r^2=9-r\ge3, \] Exercise II.7.5(d), P-AGH275AMPLEPROPS, shows that \[ M=A_r+N \] is very ample, in particular globally generated.
This induction is well-founded also at anticanonical degree zero. Indeed, if \(A_r\cdot M=0\) and every intersection \(M\cdot E\) with a \((-1)\)-curve were positive, the positive line decomposition of \(A_r\) in step 3 would force \(A_r\cdot M>0\). Thus at degree zero the zero-intersection contraction case applies until the explicit \(r=2\) base is reached.
2. The global-generation assertion of step 1 holds on\[ X_2=\operatorname{Bl}_{P_1,P_2}\PP^2. \]
Write \[ M=ah-b_1e_1-b_2e_2. \] The three \((-1)\)-curves are \[ E_1=e_1, \qquad E_2=e_2, \qquad L_{12}=h-e_1-e_2. \] The hypotheses give \[ b_1=M\cdot E_1\ge0, \qquad b_2=M\cdot E_2\ge0, \] and \[ c:=a-b_1-b_2=M\cdot L_{12}\ge0. \] Hence \[ M = c h + b_1(h-e_1) + b_2(h-e_2). \]
The class \(h\) is the pullback of the line system on \(\PP^2\), while \(h-e_i\) is the base-point-free pencil of lines through \(P_i\). Thus all three summand line bundles are globally generated. Nonnegative tensor products of globally generated line bundles are globally generated, proving the claim.
3. For every \(2\le r\le6\), the anticanonical class \(A_r\) is a positiveintegral sum of \((-1)\)-curve classes.
Write \[ L_{ij}=h-e_i-e_j. \] For \(r=2\), \[ A_2=3L_{12}+2e_1+2e_2. \] For \(r=3\), \[ A_3=L_{12}+L_{13}+L_{23}+e_1+e_2+e_3. \] For \(r=4\), \[ A_4=2L_{12}+L_{34}+e_1+e_2. \] For \(r=5\), let \[ G=2h-e_1-e_2-e_3-e_4-e_5, \] the strict transform of the conic through the five points; it is a \((-1)\)-curve. Then \[ A_5=G+L_{12}+e_1+e_2. \] Finally, \[ A_6=L_{12}+L_{34}+L_{56}. \] Expanding each right-hand side gives \[ 3h-\sum_{i=1}^r e_i=A_r. \]
4. Under the hypotheses of step 1, if moreover\[ M^2>0, \]then a general member of \(|M|\) is a nonsingular irreducible curve.
We refine the induction in step 1.
First take \(r=2\). The decomposition there writes \[ M=ch+b_1(h-e_1)+b_2(h-e_2) \] with \(c,b_1,b_2\ge0\).
If \(c>0\), the complete system \(|M|\) contains, after multiplying by a section of the remaining globally generated factor which is nonzero on a dense open, the ratios of the system \(|ch|\). The latter is the blowdown \(X_2\to\PP^2\) followed by the \(c\)-uple Veronese embedding. Hence the morphism defined by \(|M|\) is generically one-to-one, in particular separable, and has two-dimensional image.
If \(c=0\), then \[ M^2=2b_1b_2>0 \] forces \[ b_1,b_2>0. \] The two pencils \(|h-e_1|\) and \(|h-e_2|\) define a birational morphism \[ X_2\longrightarrow\PP^1\times\PP^1. \] Indeed, after taking \[ P_1=[1:0:0], \qquad P_2=[0:1:0], \] it is, on the dense chart \(z\ne0\), \[ [x:y:z] \longmapsto ([y:z],[x:z]), \] which recovers \(x/z\) and \(y/z\). The positive tensor powers \(b_1,b_2\) followed by the Segre–Veronese embedding remain generically one-to-one. Thus in this case too the morphism of \(|M|\) is separable with two-dimensional image.
Bertini’s theorem for a base-point-free separable system gives a nonsingular general member, and Bertini irreducibility for a system with two-dimensional image makes it irreducible.
Now let \(r>2\). If \(M\cdot E=0\) for some \((-1)\)-curve, step 1 writes \[ M=\sigma^*M' \] on the contraction \(\sigma:X_r\to X_{r-1}\), with \[ (M')^2=M^2>0. \] By induction a general member of \(|M'|\) is nonsingular and irreducible. Since \(M'\) is globally generated, the members passing through the contraction point form a proper hyperplane in \(|M'|\); choose the general smooth member outside that hyperplane. Its pullback is isomorphic to it and is a nonsingular irreducible member of \(|M|\).
If \(M\cdot E>0\) for every \((-1)\)-curve, step 1 proves that \(M\) is very ample. A general hyperplane section of the corresponding embedding is nonsingular and irreducible by Bertini. This completes the induction.
5. Let \(C\subseteq S\) be an irreducible curve and put\[ d=H\cdot C>0. \]Then\[ \boxed{C^2\ge d-2.} \]
Adjunction and \(K_S=-H\) give \[ 2p_a(C)-2=C^2-d. \] Because \(C\) is an integral projective curve, \[ p_a(C)=h^1(C,\OO_C)\ge0. \] Therefore \[ C^2-d\ge-2, \] which is the displayed inequality.
6. If an irreducible curve \(C\) on \(S\) has\[ C^2<0, \]then \(C\) is one of the \(27\) lines.
Step 5 gives \[ d-2\le C^2<0. \] Since \(d\) is a positive integer, this forces \[ d=1, \qquad C^2=-1. \] An irreducible degree-one curve in the given embedding \[ S\subseteq\PP^3 \] is a projective line. Thus \(C\) is one of the \(27\) lines on \(S\).
7. If an irreducible curve \(C\) is not a line and satisfies\[ C^2=0, \]then\[ \boxed{H\cdot C=2, \qquad p_a(C)=0.} \]Thus \(C\) is a conic.
Step 5 gives \[ d\le2. \] The adjunction formula becomes \[ 2p_a(C)-2=-d. \] If \(d=1\), its right-hand side is odd, impossible. Hence \[ d=2, \] and then \[ p_a(C)=0. \] In the anticanonical embedding a degree-two irreducible curve is a conic.
8. If \(C\) is an irreducible curve which is neither a line nor a conic ofself-intersection zero, then its class satisfies condition (c):\[ \boxed{C\cdot L\ge0\text{ for every line }L, \qquad C^2>0.} \]
If \(L\) is any line and \(C\ne L\), the two distinct irreducible curves have nonnegative intersection number: \[ C\cdot L\ge0. \] Since \(C\) is not a line, step 6 excludes negative self-intersection; since it is not a conic of square zero, step 7 excludes zero self-intersection. Therefore \[ C^2>0. \] This proves the necessity of alternatives (a)–(c).
9. Every class in alternative (a) contains an irreducible nonsingularcurve.
Alternative (a) is, by definition, one of the \(27\) line classes. The corresponding line \[ L\cong\PP^1 \] is itself nonsingular and irreducible.
10. Every class in alternative (b) contains an irreducible nonsingularconic.
Let \(D\) satisfy \[ H\cdot D=2, \qquad D^2=0. \] Put \[ E=H-D. \] Then \[ H\cdot E=3-2=1 \] and \[ E^2 = H^2-2H\cdot D+D^2 = 3-4 =-1. \] Also \[ K_S\cdot E=-1. \] To identify \(E\) without assuming effectivity, write \[ E=ah-\sum_{i=1}^6 b_i e_i. \] The two numerical equalities are \[ 3a-\sum_i b_i=1, \qquad a^2-\sum_i b_i^2=-1. \] Hence \[ \sum_i b_i=3a-1, \qquad \sum_i b_i^2=a^2+1. \] Cauchy–Schwarz gives \[ (3a-1)^2\le6(a^2+1), \] so \[ 3a^2-6a-5\le0. \] Thus the integer \(a\) is one of \[ 0,1,2. \] For \(a=0\), the equations force one \(b_i=-1\) and the rest zero, giving \(E=e_i\). For \(a=1\), they force exactly two \(b_i=1\), giving \(E=h-e_i-e_j\). For \(a=2\), they force exactly five \(b_i=1\), giving \[ E=2h-\sum_{j\ne i}e_j. \] These are precisely the \(27\) line classes FE-SRFCUBIC. Hence \[ E\sim L \] for a line \(L\subseteq S\), and \[ D\sim H-L. \]
The planes through \(L\) cut divisors \[ L+Q_\Pi\in|H|, \] where the residual curves \(Q_\Pi\) have class \(H-L=D\). As proved directly in P-AGH547GENUSBOUND, a general residual conic in this pencil is nonsingular and irreducible. Thus \(D\) contains such a conic.
11. Every class in alternative (c) contains an irreducible nonsingularcurve.
For the cubic surface \(S=X_6\), the lines are exactly its \((-1)\)-curves. Condition (c) says precisely that \[ D\cdot E\ge0 \] for every \((-1)\)-curve \(E\), and that \[ D^2>0. \] Step 4, applied with \(r=6\) and \(M=D\), therefore gives a nonsingular irreducible member of \(|D|\).
12. A divisor class on the cubic surface contains an irreducible curve ifand only if it contains an irreducible nonsingular curve, and this happensexactly in alternatives (a), (b), and (c).
If the class contains an irreducible curve, steps <1>6–<1>8 place it in one of the three alternatives. Conversely, steps <1>9–<1>11 show that every class in one of those alternatives contains an irreducible nonsingular curve. A nonsingular irreducible curve is in particular irreducible, so the two existence conditions are equivalent and both are equivalent to the classification (a)–(c).
13. Q.E.D.
Steps <1>1–<1>4 establish the lower-degree del Pezzo linear-system lemma suggested by the hint. Steps <1>5–<1>8 prove necessity, and steps <1>9–<1>12 prove sufficiency and the equivalence of the two existence conditions.