Homogeneous Interpolation and Some Continued Fractions

We prove: if $d/m<2280/721$, there is no curve of degree $d$ passing through $n = 10$ general points with multiplicity $m$ in $\bf{P}^2$. Similar results are given for other special values of $n$. Our bounds can be naturally written as certain palindromic continued fractions.


Introduction
Denote by L (d, n, m) the linear system of degree d curves in P 2 passing through n general points P 1 , . . . , P n with multiplicity at least m. For n ≥ 9, Nagata's conjecture ( [9]) predicts that L (d, n, m) is empty if d < √ nm. The statement is clear if n = k 2 is a square, but remains widely open otherwise. A refined conjecture due to Harbourne-Hirschowitz further predicts that in fact L (d, n, m) is non-special, i.e. of expected dimension max{−1, v}, where v = d(d + 3)/2 − nm(m + 1)/2 is the virtual dimension of L (d, n, m). We refer to ( [2], [3], [4], [5]) for background on the problem and some recent results.
The proof of the Theorem consists of two degenerations. First, we specialize the n general points P i in P 2 to general points P i on a fixed curve C of degree k. The problem is naturally reduced to an interpolation problem on a ruled surface S = P(E ) where E is a semistable rank 2 vector bundle of degree α on C. Second, we specialize the n points in S to a curve Γ of self-intersection 0 (in general, this step requires a deformation of the underlying surface S). The methods in this paper extend our previous attempt in [10]. We were greatly influenced by the work of Ciliberto-Miranda in [4].
The paper is organized as follows. In Section 2 we introduce a Basic Lemma that will be useful throughout the paper. In Section 3 we give background on ruled surfaces and elementary transforms. In Section 4 we perform the first degeneration and obtain a certain weak bound on d/m for any n ≥ 9. The construction is formalized in the next two sections. In Section 7 we sketch the proof of the Main Theorem. In each subsequent section we verify the theorem for specific values of n. In Section 12 we prove a certain refinement of the Main Theorem. In Appendix A we review the indecomposable elliptic ruled surface of degree 1. Appendix B has some auxiliary results on continued fractions.
Notation and Conventions. We work over C. Following EGA IV.4, for given a subscheme Y ⊂ X we denote by N Y /X ∼ = I Y /I 2 Y the conormal sheaf of Y . For any coherent sheaf F on X, we denote P(F ) = Proj(⊕ µ≥0 Sym µ F ).
In this section we give some background on ruled surfaces and elementary transforms. Our reference is ( [7], Ch. V). Let C be a smooth curve of genus g ≥ 0 and let S be a ruled surface over C. Let C 0 be a minimal section of S, i.e. a section of minimal self-intersection. The invariant e = C 2 0 is the degree of S (this differs by sign from Hartshorne's notation). We have: Lemma 3.1. Let E be a rank 2 vector bundle on C. Consider the ruled surface S = P(E ). The following are equivalent: (a) E is semistable; (b) e ≥ 0; (c) for any effective divisor D of S, we have D 2 ≥ 0.
If S = P(E ) with E semistable, we will also say that the surface S is semistable. By the lemma above, S is semistable if and only if S is of degree e ≥ 0.
3.1. Elementary Transforms. Let S be a ruled surface over C and let P be a point on S. We can create a new ruled surface S by applying an elementary transform at P ( [7], Example V.5.7.1). We recall the construction. Denote by F be the fiber of S through P . Let π : S → S be the blowup of P and let F be the strict transform of F in S. Finally, let π : S → S be the contraction of the (-1)-curve F in S (see Fig. 1).
Similarly, we can define elementary transforms for vector bundles. In the setting above, suppose that S = P(E ) where E is a rank 2 vector bundle on C. Consider the short exact sequence where the map on the right is just the evaluation map at P . The kernel E is again a rank 2 vector bundle on C. We can identify S = P(E ) with the surface constructed above.
The following lemma describes the behavior of semistability under elementary transforms.
Lemma 3.2. Let S = P(E ) be a semistable ruled surface. Let P be a general point on a fixed fiber F of S (the fiber F need not be general). Let S be the ruled surface obtained from S by applying an elementary transform at P . Then S is semistable, unless S ∼ = C × P 1 is the trivial ruled surface.
Proof. If S is not semistable, there exists a section C of S with (C ) 2 < 0. Denote by C be the strict transform of C in S. Since S is semistable, we have C 2 ≥ 0. It follows that C passes through P and C 2 = (C ) 2 + 1 = 0. Since P is a general point on a fiber F , it follows that S ∼ = C × P 1 .

First Degeneration
We introduce the first degeneration. The method in this section extends author's previous work in [10]. As an application we prove Theorem 4.1 below. We are unaware if the result has appeared previously in the literature in this form.
Theorem 4.1. Let n be a non-square positive integer. Write n = k 2 +α with k = √ n . Assume that either: (i) α is even, or (ii) k ≥ 3. If the linear system L (d, n, m) is nonempty, then d/m ≥ c In particular, the theorem applies for n = 3, 6 and 8 and any n ≥ 9.
Proof of the Theorem. Assume L (d, n, m) is nonempty. The idea is to specialize the n general points in P 2 to a smooth curve C of degree k and then apply Basic Lemma to estimate the multiplicity of vanishing of a general curve in L (d, n, m) along C.
Step 1. Let ∆ be the open unit disk over C and let X = P 2 × ∆. We view X as a relative plane over ∆. For any t ∈ ∆, denote the fiber X t = X × {t}. Fix a smooth curve C ⊂ X 0 of degree k. We have the following split exact sequence Let C be the section of S corresponding to the short exact sequence above. Note that Step 2. Choose any set of n distinct points P i on C (here we do not require the points P i ∈ C to be general). Next, we construct a set of n relative points P i → ∆ in X specializing to P i in a general way. Denote by P i the images of P i in S . Thus, each P i is a general point on the fiber above P i .
Let X → X be the blowup of the relative points P i and let E i denote the corresponding exceptional divisors. Denote by C the strict transform of C in X. We have the following short exact sequence: is a line bundle of degree α = n − k 2 on C. The short exact sequence corresponds to a certain element ξ ∈ Ext 1 (A, O C ). Next, consider the ruled surface S = P(N C/ X ).
We identify C with the section of S corresponding to the above exact sequence. Note that C ∼ O S (1) and  The conormal bundles N C/X and N C/ X are related by elementary transforms at the points P i : Similarly, the ruled surfaces S and S are related by elementary transforms as on Fig. 1 (the intermediate surface S will play a role later in Section 5).
Step 3. We claim: be an arbitrary element. Then ξ can be realized in the above way, for some specialization of points P i to P i .
Proof. The idea is to consider the construction in Step 2 in reversed order. Start with any extension and let S = P(E ). As before, we identify C with the section of S corresponding to the above exact sequence. Next, we construct the vector bundle E by applying elementary transforms at the points P i on C: and let P 1 , . . . , P n be as on Fig. 1. It follows that E is realized as an extension Now, the key observation is that so the above extension is trivial. This allows us to identify E ∼ = O C ⊕ O C (−kH) with N C/X , and so S with P(N C/X ). Finally, we choose the relative points P i to pass through P i in S . This identifies E with N C/ X , and so S with P(N C/ X ).
Corollary 4.5. If the specialization of P i to P i is general enough, the conormal bundle N C/ X is semistable of slope α/2. This follows from Lemma 4.4 and the following general fact: Lemma 4.6. Let C be a curve of genus g ≥ 0. Let A be a line bundle on C of degree α ≥ 0. Assume that either: (i) α is even, or (ii) g ≥ 1. Then, a general element ξ ∈ Ext 1 (A, O C ) corresponds to a semistable rank 2 vector bundle E on C.
Proof. The set of elements ξ ∈ Ext 1 (A, O C ) that correspond to semistable vector bundles E is open (this follows from [8], Thm. 2.8). So, it suffices to show that the set is nonempty. Now, if g = 0 and α = 2α 0 is even, then O P 1 (α 0 ) ⊕ O P 1 (α 0 ) is semistable. If g ≥ 1, one can prove the statement by induction on α by using Lemma 3.2. We leave this as an exercise.
Step 4. We complete the proof of the theorem. Since L (d, n, m) is nonempty by assumption, there is a flat family of curves C → ∆, where C is a nontrivial section of |O X (dH − mE i )|. We are interested in estimating µ = mult C/ X ( C ), which of course is the same ( ) By Lemma 4.5 and Cor. 2.2, we have: Combining ( ) and ( ), we get: One can easily check that this is equivalent to the inequality in the theorem. See also Lemma B.1(a) in the Appendix.

Reduction of Interpolation Problems to Ruled Surfaces
We formalize some results from the previous section. Our result here is Theorem 5.5 which will be used through the rest of the paper.
Notation 5.1. A marked surface (S; P 1 , . . . , P n ) is simply a surface S together with n distinct points P 1 , . . . , P n on S.
Notation 5.2. Let S = (P(E ); P 1 , . . . , P n ) be a marked ruled surface over C. For any integers (µ, b, m) and a line bundle b of degree b on C, we denote the line bundle on the blowup π : S → S at the points P 1 , . . . , P n , with exceptional divisors e 1 , . . . , e n . Here we denote O S (µ) = π * O S (µ).
where α = deg(E ) and g is the genus of C.
Notation 5.4. Let C be a smooth curve, A a line bundle on C and let ξ ∈ Ext 1 (A, O C ) corresponding to an extension 0 → O C → E → A → 0. We denote by S(C, A, ξ) the ruled surface S = P(E ) and we identify C with the section determined by the short exact sequence. Note that The following theorem allows to reduce interpolation problems on P 2 to certain interpolation problems on ruled surfaces.
Theorem 5.5. Let n be a non-square positive integer. Write n = k 2 + α with k ≥ 1 and α ≥ 0. Consider the linear system L (d, n, m) for some positive integers d and m. Fix a smooth curve C of degree k in P 2 . Let S = S(C, A, ξ; {P i }) be a marked ruled surface where: • A is any line bundle of degree α on C; • ξ ∈ Ext 1 (A, O C ) is any element; • P 1 , . . . , P n are distinct points on C ⊂ S such that P i ∼ A + kH. Then, for any µ, we have where: The following lemma justifies our definition of L S (µ, b, m): Lemma 5.6. In the setting of the theorem, we have Proof. This is an easy computation. We have: The lemma follows.
Proof of Theorem. We will use the same construction as in the proof of Theorem 4.1.
Step 1. Consider the threefold X = P 2 × ∆. We identify C with a curve on X 0 . Given ξ ∈ Ext 1 (A, O C ), we specialize the n relative points P i to P i ∈ C as in Lemma 4.4. As before, let X → X be the blowup of the P i , and let C be the strict transform of C in X. We have the short exact sequence By construction, the above extension corresponds to ξ. In is the ruled surface we started with.
Step 2. Consider the threefold Y obtained from X = P 2 × ∆ by first blowing up C (with exceptional divisor S = P(N C/X )), followed by blowing up the strict transforms of the relative points P 1 , . . . , P n (with exceptional divisors E 1 , . . . , E n ). We view Y → ∆ as a flat family with general fiber Y t ∼ = X t . The special fiber Y 0 is the union of two surfaces S ∪ X 0 meeting transversely along C ∼ = C. 1 This construction is related to the construction in Step 1 as follows. Let X be the threefold obtained from X by blowing up C (with exceptional divisor S = P(N C/ X )). Then, Y can be obtained from X by applying (-1)-transfers to the exceptional curves e 1 , . . . , e n as on Fig. 2 (see also [3], Section 4.1). The induced map π : S → S coincides with the corresponding map on Fig. 1. Step 3. For a given µ, consider the following line bundle on Y : We view L Y as a flat family of line bundles with general fiber L Yt ∼ = O Xt (dH − mE i ). The special fiber L Y0 is described by the following short exact sequence: Lemma 5.7. We have: Proof. The line bundle L S has the following properties: To complete the proof of the theorem, take cohomology in ( * ): Hence, the coboundary map δ = 0. The theorem now follows from the semicontinuity principle applied to h 0 (L Yt ).
Corollary 5.8. Assume the above setting.
(a) For any µ, we have: Proof. (a) This follows from the short exact sequence ( * ) and the fact that χ(L Yt ) is a constant function of t.
(b) This follows from (a).
Proof. Consider the long exact cohomology sequence associated to ( * ).
is an isomorphism. The claim follows.

Families of Ruled Surfaces
We can use the degeneration technique from the previous section to reduce an interpolation problem on P 2 to an interpolation problem on a certain ruled surface S. We would like to perform further degenerations to study the later problem. Our first goal is to define an object S(Z, A , ξ) → ∆ which is a relative analogue of S(C, A, ξ). We conclude with a technical result (Prop. 6.4) which will be used in Sections 10 and 13. As usual, ∆ denotes the open unit disk over C.
such that E is locally free. We denote by S(Z, A , ξ) → ∆ the relative ruled surface S = P(E ) → ∆ together with the subscheme Z = P(A ) defined by the above exact sequence. In particular, Z is a divisor of S with Z ∼ O S (1).
We will assume that W is supported on C × {0}. In applications, W will be reduced; however, everything we say in this section holds in the more general setting. Lemma 6.2. In the above setting, the projection p : Proof. Since A is invertible, Z = P(A ) = P(I W ) which is exactly the definition of a blowup. The rest is clear.
Consider the following question: given A as above, which elements ξ ∈ Ext 1 (A , O C×∆ ) correspond to a locally free extension E ? Clearly, if W = ∅, then A is locally free and so any ξ will do. In the general case, we have: Proof. See [6], Chapter 2, p. 36-37. The exact sequence follows from the local-to-global spectral sequence Consider a relative ruled surface S(Z, A , ξ) → ∆. For any t ∈ ∆, S t = P(E t ) is a ruled surface over C × {t} where E t arises as an extension . For a general t, the subscheme Z t is a section of S t . On the special fiber, we have Z 0 = C 0 ∪ F 0 , where C 0 is a section of S 0 and F 0 = F × {0} is the vertical component of Z 0 . If F 0 = ∅, we will say that Z 0 is a degenerate section of S 0 .
Next, we will show that any degenerate section can be smoothed, in the following sense.
be any extension, with E 0 locally free. Then, the exact sequence can be extended to Proof. Consider the commutative diagram with exact rows: It follows that the bottom row of the diagram splits. Now, the map on the left factors through which is surjective. The map on the right is just the restriction which is also surjective. By the Short Five Lemma, the map in the middle is surjective as well. Hence, any given extension ξ 0 ∈ Ext 1 (A 0 , O C×{0} ) can be lifted to ξ ∈ Ext 1 (A , O C×∆ ). The resulting E is locally free by Prop. 6.3(b).

Main Result -Overview
The following theorem was announced in the introduction. The proof will occupy the rest of the paper. We will consider a certain refinement in Section 12.
If the linear system L (d, n, m) is nonempty, then d/m ≥ c n .
The proof of the theorem consists of the four steps outlined below.
7.1. Setup. We assume L (d, n, m) is nonempty. Fix a smooth curve C of degree k in P 2 . By Cor. 5.9, the linear system |L S (µ, b, m)| is nonempty, with µ = d/k , for any marked ruled surface S(C, A, ξ; {P i }) as in Theorem 5.5.

7.2.
Degeneration. We will construct a relative marked ruled surface S(Z, A , ξ; {P i }) over ∆ such that the general fiber S t satisfies the assumptions of Theorem 5.5, i.e.: • A is a line bundle on C × ∆ of relative degree α = deg A t .
• P 1 , . . . , P n lie on Z. We denote by P i the projection of P i under Z → C × ∆.
• For t ∈ ∆ general, we have P i,t ∼ A t + kH on Z t ∼ = C. We now describe the special fiber S 0 . We assume that S 0 is semistable. Next, we assume that there is a smooth (possibly disconnected) curve Γ on S 0 with the following two properties: • Γ meets Z 0 transversely at n distinct points points P i = P i,0 ; • Γ 2 = 0. This determines uniquely the numerical class of Γ: . In particular, a necessary condition for the existence of Γ is that α | 2n. Let Γ = s i=1 Γ i where each Γ i is a smooth irreducible curve, with Γ i · Γ j = 0 for i = j. Since Γ 2 = 0 and S is semistable, Γ lies on the boundary of the effective cone of S 0 (this follows from Lemma 3.2). Therefore, Γ i ≡ λ i Γ for some λ i ∈ Q with λ i = 1 (in fact, in applications we will always have λ 1 = · · · = λ s = 1/s). 7.3. Semistability. Denote by π : S → S the blowup of P 1 , . . . , P n and let E 1 , . . . , E n be the corresponding exceptional divisors. Denote by Γ the strict transform of Γ. We make the following hypothesis: • for each i, the conormal bundle N Γi/ S is semistable of slope 1 2 λ i n.

7.4.
Invariants. Assuming the construction above can be realized, we complete the proof of the theorem. Consider the line bundle By construction, there is a flat family of curves C → ∆ in S, where C is a section of |L S (µ, B, m)|. Denote by C the projection of C in S. The following is a key computation: Proof. Using the fact that Γ · Z 0 = n, Γ · f = 2n α and C 0 ∼ µZ 0 − bf , we find: Finally, substitute b = nm − kd and n = k 2 + α.
To complete the proof of the theorem, we will estimate µ i = mult Γi/S (C ) in two ways. First, there is an obvious upper bound which comes from the numerical class of Γ: Since µ ≤ d/k, this becomes: By Basic Lemma and the semistability hypothesis, we have: From ( ) and ( ), and using λ i = 1, we get: It turns out that this is equivalent to the inequality in the theorem. We will check this explicitly for specific values of n. For the general case, see Lemma B.1(b) in the Appendix.

Eight Points
We verify Main Theorem in the case n = 8. The value c (2) 8 = 48 17 is well-known to be sharp (see example below). We include this case for illustration purposes. 8.1. Setup. Since k = 2, we take C to be a smooth conic in P 2 . The line bundle A ∼ = O P 1 (4) on C is of degree α = 4. Consider an extension corresponding to a general ξ ∈ Ext 1 (A, O C ). It follows that Hence, S = P(E ) ∼ = C × P 1 ∼ = P 1 × P 1 . We identify C with the section of S corresponding to the short exact sequence. It follows that C ∼ C 0 + 2f where C 0 is a horizontal section of S.

8.2.
Degeneration. Let S = S × ∆ and Z = C × ∆ ⊂ S. We identify S with the special fiber of S → ∆. We take Γ = Γ 1 + · · · + Γ 4 on S, where each Γ i ∼ C 0 is a general horizontal section of S. Let Γ i ∩ C = {P 2i−1 , P 2i }. Next we specialize the eight relative points P i ⊂ Z to P i ∈ C in a general way. 8.3. Semistability. Let S be the blowup of S along the relative points P i . We have to show that, for each i, the conormal bundle N Γi/ S is semistable (of slope 1). Denote by P i the image of P i in the corresponding P(N Γ/S ) ∼ = Γ × P 1 . Then, P(N Γi/ S ) is obtained from P(N Γi/S ) by performing elementary transforms at the points P 2i−1 , P 2i . If the specialization is general enough, the points P 2i−1 , P 2i do not belong to the same horizontal section of P(N Γi/S ). It follows that , which is semistable of slope 1.

Ten Points
Here we prove Main Theorem for n = 10 points.
9.1. Setup. Since k = 3, we take C to be a smooth cubic in P 2 . Fix a point W on C such that 9W ∼ 3H.
For example, we can take W to be a Weierstrass point of C (however, later in Section 12 we will require that 3W H). Let A = O C (W ) which is of degree α = 1. Since h 1 (A ∨ ) = 1, there is a unique nontrivial extension The surface S = P(E ) is an indecomposable elliptic ruled surface of degree 1 (see Appendix A for background). The short exact sequence determines a minimal section of S which we identify with the curve C.
Finally, we specialize the ten relative points P 1 , . . . , P 10 in Z to P 1 , . . . , P 10 in a general way such that for any t ∈ ∆.
9.3. Semistability. We claim that, for each i, N Γi/ S is semistable (of slope 1). Denote by P i the image of P i in P(N Γ/S ) ∼ = P(O Γ ⊕ O Γ ) ∼ = Γ × P 1 . Now, P(N Γi/ S ) is obtained from P(N Γi/S ) by performing elementary transforms at the points P 2i−1 , P 2i . If the specialization is general enough, the points P 2i−1 , P 2i do not belong to the same horizontal section of P(N Γi/S ). It follows that 9.4. Invariants. Denote µ i = mult Γi/S (C ). By symmetry, µ 1 = · · · = µ 5 . Since Γ i ≡ 4C − 2f , we have the following upper bound: ( ) The lower bound from Basic Lemma is: From ( ) and ( ), we get: q.e.d.

Eleven Points
We prove Main Theorem for n = 11 points. This is the first time when we study an interpolation problem L S (µ, b, m) by deforming the underlying surface S itself.
10.1. Setup. As before, C is a smooth cubic in P 2 . Fix a point W on C such that 9W ∼ 3H. Let A be any line bundle of degree α = 2 on C. It is easy to see that for a general ξ ∈ Ext 1 (A, O C ), the ruled surface S(C, A, ξ) is decomposable of degree 0. Denote by C (i) , i = 0, 1, the two minimal sections of S. It follows that C ≡ C (i) + f .

10.2.
Degeneration. We will construct a relative marked ruled surface S(Z, A , ξ; {P ij }) such that: • The special fiber S 0 is simply C × P 1 .
• The special section Z 0 = C 0 ∪ F ; here C 0 is a horizontal section of S 0 and F is the fiber of The construction is done as follows. First, we choose relative points P i in C × ∆ specializing to W × {0} in a general way. Let Since 9W ∼ 3H, it follows that Consider the following short exact sequence on C × {0}: By Prop. 6.4, the sequence can be extended to with exceptional divisor F . Finally, we take P i to be the strict transform of P i in Z. Figure 4. The blowup Z → C × ∆ 10.3. Semistability. We take Γ = Γ 1 + · · · + Γ 11 where Γ i is the horizontal section of S 0 = C × P 1 through P i (see Fig. 4). Consider the blowup S → S at the relative points P i . We claim that for each i = 1, . . . , 11, N Γi/ S is indecomposable of degree 1 (hence semistable of slope 1/2). First, we will show that N Γi/S is indecomposable of degree 0. We will need some deformation theory. Let D = C[t]/t 2 be the ring of dual numbers. Let S = S × ∆ D viewed as an infinitesimal deformation of S 0 ∼ = C × P 1 over D. We will say that a section T of S 0 is (infinitesimally) unobstructed if and only if T can be extended to a subscheme T of S flat over D.
is a line bundle on C = C × D. By Nakayama's lemma, for any i = j, the induced map E → L i ⊕ L j is an isomorphism. Hence Now consider the short exact sequence from part (a): Part (c) of the lemma implies that S is not an infinitesimally trivial deformation. By part (b) and by symmetry, Γ i is obstructed for any i. It follows that the conormal bundle N Γi/S is indecomposable of degree 0.
Denote by P i the image of P i in P(N Γi/S ). Clearly, P i does not belong to the unique minimal section of P(N Γi/S ) (because P i meets S 0 transversely). Finally, P(N Γi/ S ) is obtained from P(N Γi/S ) by performing an elementary transform at P i . It follows that N Γi/ S is indecomposable of degree 1.
Remark 10.2. It might be also profitable to study the behavior of C along the fiber F . We have N F / S ∼ = O P 1 (10) ⊕ O P 1 (1), which follows from the split exact sequence (10) and N F / Z ∼ = N F/Z ∼ = O P 1 (1). The fact that N F / S is unstable causes certain multiplicity and tangency conditions on the limit curve C 0 at the point W = C 0 ∩ F . A more careful analysis of the situation is beyond of the scope of this paper.

Twelve Points
In this section we prove Main Theorem for n = 12. This case is similar to n = 10.
11.1. Setup. As before, C is a smooth cubic in P 2 . Fix a point W on C such that 9W ∼ 3H. Take A = O C (3W ) which is of degree α = 3. It is easy to see that for a general ξ ∈ Ext 1 (A, O C ), S(C, A, ξ) is an indecomposable elliptic ruled surface of degree 1. It follows that C ≡ C 0 + f where C 0 is a minimal section of S.

11.2.
Degeneration. Let S = S × ∆ and Z = C × ∆ ⊂ S. We identify S with the special fiber of S. Take Γ = Γ 1 + Γ 2 where each Γ i is a general section of the pencil | − 2K S |. In particular, . . , P 6 } and C ∩ Γ 2 = {P 7 , . . . , P 12 }. It follows that Next, we specialize P i in Z to P i in a general way such that for any t ∈ ∆.
11.3. Semistability. We have to show that if the specialization of the points P i is general enough, the conormal bundle N Γi/ S is semistable (of slope 3). Since semistability is an open property ( [8], Thm. 2.8), it suffices to describe a particular specialization for which N Γi/ S is semistable. This is not hard. In fact, we claim that we can specialize the points in such a way that and similarly for Γ 2 . This can be achieved by moving the triples of points {P 3i−2 , P 3i−1 , P 3i } "in parallel" while being assigned to the same section of the pencil | − 2K S |. Figure 5. Specialization to Γ (n = 12) 11.4. Invariants. Denote µ i = mult Γi/S (C ). By symmetry, µ 1 = µ 2 . Since Γ i ≡ 4C 0 − 2f , we have the following upper bound: ( ) The lower bound from Basic Lemma is: Combining ( ) and ( ), we get: This completes the proof for twelve points.

A Refinement
Here we prove a certain refinement of the Main Theorem in the case of n = 10, 11 and 12 points. We work in the setting of the previous sections. The idea is to show that, under some additional assumptions, the inequality ( ) can be replaced by a stronger inequality ( ). First, we have: Lemma 12.1. Let n = 10, 11 or 12. Consider the degeneration described above for the particular value of n. Let b : Since 9W ∼ 3H, it follows that 3b ∼ 3bW .
(b) Assume 1 2 µ 1 = γ. By Basic Lemma, there is an injective morphism: Now the idea is to show that the vector bundle on the right hand side decomposes as a direct sum of line bundles of the same degree as O Γ1 (− C ). It will follow that O Γ1 (− C ) is isomorphic to one of the summands. Below we consider each case for n separately.
Recall that N Γ1/ S is indecomposable of degree 1 (determinant W ). From the results in Appendix A, is isomorphic to one of the summands. It follows that 2b ∼ 2bW . Since 3b ∼ 3bW and gcd(2, 3) = 1, we conclude that b ∼ bW . Case n=12. This is similar to the case of ten points. We leave the details to the reader.
The following result is a refinement of the Main Theorem. Note that it only applies when 3 d.
Proposition 12.2. Let n = 10, 11 or 12. If L (d, n, m) is nonempty and 3 d, then κ n ≥ 0 with Proof. Fix W so that 9W ∼ 3H but 3W H. Since 3 d, part (a) of the lemma implies that b bW . From part (b), we get: Finally, ( ) together with ( ) imply the desired inequality.

The Remaining Case
Here we prove the Main Theorem in the case when k ≥ 3, α is even, α | 2n. This generalizes the case of eleven points (in fact, the proof can be also applied in the case of eight points).
13.1. Setup. Let C be a smooth plane curve of degree k. We will make the following assumption: there is a divisor W = W 1 + · · · + W α 2 on C, where W i 's are distinct points, such that 2k 2 α W ∼ kH. Here is one way to construct such a curve. Fix a line ⊂ P 2 and let W 1 , . . . , W α 2 be distinct points on . Now, take C to be any smooth curve of degree k which is tangent to to order 2k α at each of the points W i . It follows that 2k α W ∼ H, which satisfies the assumption. 13.2. First Degeneration. It will be convenient to re-index the n relative points as {P ij } where i = 1, . . . , 2n α and j = 1, . . . , α 2 . We will construct a relative marked ruled surface S(Z, A , ξ; {P ij }) such that: • The special fiber S 0 is simply C × P 1 .
• The special section Z 0 = C 0 ∪ F 1 ∪ · · · ∪ F α 2 ; here C 0 is a horizontal section of S 0 and F j is the fiber of S 0 above W j , for each j.
• The relative points {P ij } on Z are such that P ij,t ∼ A t + kH on Z t ∼ = C, for general t. • Each limit point P ij = P ij,0 is a general point on the fiber F j . The construction generalizes the case of eleven points. Namely, we first choose relative points P ij in C × ∆ specializing to W j × {0} in a general way. Next, let A = O C×∆ ( P ij − kH) and A = A ⊗ I W ×{0} . It follows that Consider the short exact sequence on C × {0}: (Note that h 0 (C, O C (W )) = 1, so the sequence is unique). By Prop. 6.4, the sequence can be extended to where E is locally free. Finally, we take S = P(E ) and Z = P(A ). It follows that We take P ij to be the strict transform of the P ij on the blowup Z → C × ∆ at W × {0}.
13.3. Semistability (α = 2). Let Γ i be the horizontal section of S 0 through P i,1 . Just as in the case of eleven points, we can show that the conormal bundle N Γi/ S is semistable of slope 1/2.
13.5. Semistability (α ≥ 4). Denote by S × ∆ the blowup of S × ∆ at the relative points P ij constructed in the previous step. Now, P(N Γi/ S×∆ ) is obtained from P(N Γi/S×∆ ) = P(O C ⊕ O C ) by applying α/2 elementary transforms at general points on the fixed fibers through W 1 , . . . , W α 2 . Since α/2 ≥ 2, it follows that the resulting vector bundle is semistable of slope α/4 (the proof is similar to that of Lemma 3.2). 13.6. Invariants. The computation of invariants was carried out in Section 7. This completes the proof of the Main Theorem.
Appendix A. The Indecomposable Elliptic Ruled Surface of Degree 1 Below we summarize some facts about the indecomposable elliptic ruled surface of degree 1. Our references are [1] and ( [7], Chapter V.2).
Let C be an elliptic curve. Let E be an indecomposable rank 2 vector bundle of degree 1 on C. Then, E arises as the unique nontrivial extension where A = det(E ).
Let us compute the symmetric powers of E . By ( [1], Lemma 22 on p.439), we have: where the L i are the nontrivial line bundles with L ⊗2 i ∼ = O C . Also, by ( [1], Cor. to Thm. 7 on p.434), we have E ⊗ L i ∼ = E and E * ∼ = E ⊗ A −1 . Finally, we have the Clebsch-Gordan formula ([1], p.438) for a rank 2 vector bundle: Using the above, we find: In general, if m = 2k is even, Sym m E decomposes as a sum of line bundles that are isomorphic to A ⊗k or A ⊗k ⊗ L i . If m = 2k + 1 is odd, Sym m E decomposes as a sum of k + 1 copies of A ⊗k ⊗ E .
Next, consider the ruled surface S = P(E ) together with the projection π : S → C. We identify C with the unique section of |O S (1)|. The anticanonical class of S is −K S ∼ 2C − π * (A).
Note that K 2 S = 0. We have: Proposition A.1. Let E be the indecomposable vector bundle of rank 2 and degree 1, det(E ) = A. Consider the ruled surface S = P(E ).