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Quasi-Homogeneous Linear Systems on P^2 with Base Points of Multiplicity 6

QUASI-HOMOGENEOUS LINEAR SYSTEMS ON P2 WITH BASE POINTS OF MULTIPLICITY 6
MICHAEL KUNTE

arXiv:math/0404169v1 [math.AG] 7 Apr 2004

Abstract. In this paper we prove the Harbourne-Hirschowitz conjecture for quasihomogeneous linear systems of multiplicity 6 on P2 . For the proof we use the degeneration of the plane by Ciliberto and Miranda and results by Laface, Seibert, Ugaglia and Yang. As an application we derive a classi?cation of the special systems of multiplicity 6.

1. Introduction A classical problem in algebraic geometry is the dimensionality problem for plane curves, which can be formulated as follows. Given ?nitely many general points of the projective plane with assigned multiplicities and a number d, determine the dimension of the linear system of curves of degree d having at the given points at least the assigned multiplicities. More precisely, the problem is to classify all systems which fail to have the expected dimension (see [C00] for some remarks on the history of this problem and its geometric meaning). Harbourne and Hirschowitz conjecture that these special systems are precisely the (?1)-special systems. In this paper, we give a complete list of the (?1)-special systems in the case in which the assigned multiplicity is 6 at all but one of the given points. Our main result is the proof of the Harbourne-Hirschowitz conjecture in this case. We proceed along the following lines. In Section 2 we introduce the necessary notation and give a precise statement of the Harbourne-Hirschowitz conjecture. In Section 3 we present a list of the (?1)-special linear systems in our case. Its completeness is proved in Section 4. In Section 5 we review the degeneration of the plane by Ciliberto and Miranda. This method is the key tool in our proof of the main result which is given in the ?nal two sections. 2. The Harbourne-Hirschowitz conjecture We work over the complex numbers and choose n + 1 general points p0 , p1 , . . . , pn in P2 , the projective plane over that ?eld. 2.1 Notation We write L = L(d, m0 , m1 , . . . , mn ) ? P(Γ(P2 , OP2 (d))) for the linear system of all curves of degree d in P2 having multiplicity at least mi at pi for all i. We denote by ?(L) its projective dimension. Let P′ be the blow-up of P2 at p0 , p1 , . . . , pn . By H we denote the pull-back of a line in P2 and by Ei the exceptional divisor over pi . The dimension of L is the same as the dimension of |D| on P′ with D = dH ? m0 E0 ? m1 E1 ? . . . ? mn En . Using cohomology on P′ , we have ?(L) = h0 (OP′ (D)) ? 1. Therefore we have by Riemann-Roch ?(L) = D.(D ? KP′ ) + h1 (OP′ (D)) ? h2 (OP′ (D)) + χ(OP′ ) ? 1 2
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(KP′ denotes the canonical divisor on P′ ). Since the arithmetic genus of P′ is zero, Serre duality implies D.(D ? KP′ ) ?(L) = + h1 (OP′ (D)). 2 2.2 De?nition We de?ne the virtual dimension v(L) of L as follows: v(L) = We de?ne the expected dimension to be e(L) = max{?1, v(L)}. As v(L) = d(d+3) ? n mi (mi +1) , one sees that the expected dimension is the one we i=0 2 2 obtain if all conditions imposed on the base points are independent. We de?ne L to be special or non-regular if ?(L) > e(L), otherwise we call L non-special or regular. We recall some de?nitions from [CM98]: 2.3 De?nition ((?1)-special systems) ? Let A in P2 be an irreducible curve such that its strict transform A in P′ is rational and smooth. Then A is a (-1)-curve if the self-intersection number ? A2 = ?1. ? By L.A we denote the intersection number D.A on P′ . The linear system L is called (-1)-special if ? there exist A1 , . . . , At (?1)-curves with L.Ai = ?ni such that ni ≥ 1 for all i, ? there is an j with nj ≥ 2 and ? the residual system M = L ? t ni Ai has v(M) ≥ 0. i=0 The main conjecture can be formulated as follows: 2.4 Conjecture (Harbourne-Hirschowitz) A linear system L = L(d, m0 , m1 , . . . , mn ) is special if and only if it is (?1)-special. It is easy to see that a (?1)-special system L is special because L.(L ? KP′ ) (M + nA).(M + nA ? KP′ ) = . 2 2 ? = ?1 by the rationality of A, this implies v(L) = v(L) = v(M) + D.(D ? KP′ ) . 2

Since A.KP′

?n2 + n ?n2 + n ≤ ?(L) + . 2 2 Therefore the opposite direction of the Harbourne-Hirschowitz conjecture is the non-trivial one. It states that every special system L has ?xed multiple (?1)-curves. Proving the conjecture leads to an answer of the dimensionality problem. 2.5 Remark We give a list of results on the conjecture. In fact we use all of them in several ways for the proof of our main theorem. We write L = L(d, mb0 , mb1 , . . . , mbr ) if L has precisely bi base points of multiplicity mi r 0 1 for i = 0, . . . , r. With this notation the conjecture holds if

QUASI-HOMOGENEOUS LINEAR SYSTEMS ON P WITH BASE POINTS OF MULTIPLICITY 6

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? ? ? ? ? ?

b0 + . . . + br ≤ 9 [H89], L = L(d, mn ) (call it homogeneous of multiplicity m) and m ≤ 12 [CM00], L = L(d, m0 , mn ) (call it quasi-homogeneous of multiplicity m) and m ≤ 3 [CM98], L = L(d, m0 , 4n ) [S99] and [L99], L = L(d, m0 , 5n ) [LU02] or all multiplicities are bounded by 6, i.e. mi ≤ 6 for i = 0, 1, . . . , n [Y03]. 3. Main Results

Our main result is a proof of the Harbourne-Hirschowitz conjecture in the quasihomogeneous case of multiplicity 6: Theorem A (Main Theorem) A system L(d, m0 , 6n ) is special if and only if it is (?1)-special. We give the proof within an extra section. For the proof we need the following classi?cation: Theorem B (Classi?cation of (?1)-special systems L(d, m0 , 6n )) The following is a complete list of all (?1)-special systems L(d, m0 , 6n ). d ? m0 0 1 2 3 system L(d, d, 6n ) L(d, d ? 1, 6n ) L(10e, 10e ? 2, 62e ) L(d, d ? 2, 6n ) L(9e, 9e ? 3, 62e ) L(9e + 1, 9e ? 2, 62e ) L(d, d ? 3, 6n ) L(8e, 8e ? L(8e + 1, 8e ? 3, 62e ) L(8e + 2, 8e ? 2, 62e ) L(d, d ? 4, 6n ) L(7e, 7e ? 5, 62e ) L(7e + 1, 7e ? 4, 62e ) L(7e + 2, 7e ? 3, 62e ) L(7e + 3, 7e ? 2, 62e ) L(6e, 6e ? 6, 62e ) 4, 62e ) v(L) ?21n + d ?21n + 2d ?12e ? 1 ?21n + 3d ? 1 ?6e ? 3 ?6e + 1 ?21n + 4d ? 3 ?2e ? 6 ?2e ? 1 ?2e + 4 ?21n + 5d ? 6 ?10 ?4 2 8 ?15 ?(L) ?6n + d ?11n + 2d 0 ?15n + 3d ? 1 0 2 ≥ ?18n + 4d ? 3
= if d =
9n 2

d ≥ 6n ≥ 6 d ≥ 11 n ≥ 11 2 2 e≥1 d ≥ 1+15n ≥ 16 3 3 e≥1 e≥1 d ≥ 18n+3 ≥ 21 4 4 e≥1 e≥1 e≥1 d ≥ 20n+6 ≥ 5 e≥1 e≥1 e≥1 e≥1 e≥1

+ 1 or n odd

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0 2 5 ≥ ?20n + 5d ? 6
= if d = 4n + 2 or n odd

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d ? m0

system L(6e + 1, 6e ? 5, 62e ) L(6e + 2, 6e ? 4, 62e ) L(6e + 3, 6e ? 3, 62e ) L(6e + 4, 6e ? 2, 62e ) L(5e + 2, 5e ? 5, 62e ) L(5e + 3, 5e ? 4, 62e ) L(5e + 4, 5e ? 3, 62e ) L(5e + 5, 5e ? 2, 62e ) L(4e + 4, 4e ? 4, 62e ) L(4e + 5, 4e ? 3, 62e ) L(4e + 6, 4e ? 2, 62e ) L(10, 2, 63 ) L(24, 16, 69 ) L(3e + 6, 3e ? 3, 62e ) L(3e + 7, 3e ? 2, 62e ) L(9, 0, 63 ) L(10, 1, 63 ) L(14, 5, 65 ) L(18, 9, 67 ) L(2e + 8, 2e ? 2, 62e ) L(10, 0, 63 ) L(14, 4, 65 ) L(13, 2, 65 ) L(14, 3, 65 ) L(12, 0, 65 ) L(13, 1, 65 ) L(14, 2, 65 ) L(13, 0, 65 ) L(14, 1, 65 ) L(14, 0, 65 )

v(L) ?8 ?1 6 13 ?2e ? 5 ?2e + 3 ?2e + 11 ?2e + 19 ?6e + 8 ?6e + 17 ?6e + 26 ?1 ?1 ?12e + 24 ?12e + 34 ?9 1 ?1 ?3 ?20e + 43 2 4 ?4 8 ?15 ?2 11 ?1 13 14

?(L) 2 5 9 14 ?2e + 5 ?2e + 9 ?2e + 14 ?2e + 20 ?6e + 14 ?6e + 20 ?6e + 27 2 0 ?12e + 27 ?12e + 35 0 4 0 0 ?20e + 44 5 5 2 9 0 4 12 5 14 15 e≥1 e≥1 e≥1 e≥1 2≥e≥1 4≥e≥1 7≥e≥1 10 ≥ e ≥ 1 2≥e≥1 2≥e≥1 4≥e≥1

7

8

9

2≥e≥1 2≥e≥1

10

2≥e≥1

11 12

13 14

4. The Classification In the paper [CM98] of Ciliberto and Miranda a lot of classi?cation work has been done which we can apply to our problem. Ciliberto and Miranda introduced two notions which we recall now to use their results. Let L be a linear system of plane curves with general multiple base points as above. Then L is a quasi-homogeneous (-1)-class if L = L(d, m0 , mn ), on P′ the self-intersection number L.L = ?1 and the arithmetic genus gL = L2 + L.KP′ + 1 = 0. 2

As v(L) = L2 ? gL + 1, these systems are never empty. In this case, if A is a (?1)-curve such that A ∈ L then by L.A = ?1 and the irreducibility of A, we have L = {A}. So we can identify (?1)-curves and quasi-homogeneous (?1)classes and write A = L. Ciliberto and Miranda proved that such a (?1)-curve exists up to m ≤ 6. Hence a numerical classi?cation of these systems gives a classi?cation for all

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quasi-homogeneous (?1)-curves up to multiplicity m = 6. Such a classi?cation is given in [CM98]. Now we consider the following phenomenon: Let L = L(d, m0 , mn ) be a quasihomogeneous linear system and A a (?1)-curve such that A = L(δ, ?0 , ?1 , . . . , ?n ) and L.A ≤ ?2. Let Permn be the permutation group on n letters and let σ ∈ Permn . We de?ne Aσ = L(δ, ?0 , ?σ(1) , . . . , ?σ(n) ). Then, as A is a (?1)-curve, it follows that Aσ is again a (?1)-curve. As L is quasi-homogeneous we have again L.Aσ ≤ ?2. Therefore we can construct a composition of (?1)-curves, which split o? the system L. We de?ne the set A ? Permn to be maximal such that all Aσ with σ ∈ A are pairwise di?erent. Then we de?ne a new plane curve Atot = σ∈A Aσ (see [LU02]). We call a linear system L′ = L(d, mo , m1 , . . . , mn ) as above a quasi-homogeneous (-1)con?guration if Atot is a generic element in L′ . We note that L′ is by construction quasihomogeneous (if k = |A| then there exists a ?′ such that L′ = L(kδ, k?0 , ?′n )). 4.1 Lemma (splitting-o? Lemma) Let L = L(d, m0 , mn ). Then every (?1)-curve A with L.A ≤ ?2 is of one of the following types (We have listed the associated quasi-homogeneous compound (?1)-con?gurations, too.): A = L(δ, ?0 , ?n ) 1 n?1 A = L(δ, ?0 , ?2 ? 1, ?2 ) Atot = L(nδ, n?0 , (n?2 ? 1)n ) n?1 A = L(δ, ?0 , ?2 + 1, ?2 ) Atot = L(nδ, n?0 , (n?2 + 1)n )

Proof: First one proves that strict transforms of di?erent Aσ = Aσ′ cannot meet positively on P′ . This is the case as otherwise one sees, by the Riemann-Roch theorem on P′ , that the sum of these moves in a linear system of positive dimension, which is a contradiction to being a ?xed part of L. This implies that all the di?erent Aσ are linearly independent in Pic(P′ ). Let the ?1 , . . . , ?n occur in sets of size k1 ≤ . . . ≤ ks . As rank Pic(P′ ) = n + 2 we see by n! combinatorial reasons that for the k1 !···ks ! di?erent (?1)-curves Aσ only the possibilities s = 1, k1 = n or s = 2, k1 = 1, k2 = n ? 1 can occur. That means we have at most three di?erent multiplicities ?0 , ?1 and ?2 . Moreover we have the equations A.A = ?1 and A.Aσ = 0 on P′ . That gives A.A?A.Aσ = ?1 which is equivalent to (?1 ? ?2 )2 = 1 (see [CM98]).

For the purpose of classifying the systems L(d, m0 , 6n ) we need a complete list of all (?1)curves which might split o? such systems two times. These (?1)-curves can not have higher multiplicities than 3 at the points p1 , . . . , pn . We obtain the following result: 4.2 Lemma (classi?cation of (?1)-curves) All (?1)-curves A and quasi-homogeneous (?1)-con?gurations Atot up to multiplicity 3 in the points p1 , . . . , pn which might split o? a quasi-homogeneous system L = L(d, m0 , 6n ) are elements of the systems in the following list (see [LU02]):

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not compound L(2, 0, 15 ) L(e, e ? 1, 12e ) L(1, 1, 11 ) L(1, 0, 12 ) L(6, 3, 27 ) L(12, 8, 39 )

compound e≥1 L(n, n, 1n ) L(3, 0, 23 ) n≥2

In particular, all the (?1)-curves are quasi-homogeneous. Proof: We refer to [CM98, Example 5.1] for the proof of a list of all quasi-homogeneous (?1)classes up to multiplicity 3. In [CM98, Example 5.15] is given a complete list of all quasihomogeneous (?1)-con?gurations up to multiplicity 3. Using this two lists and Lemma 4.1 gives this result. Now we give the proof of the classi?cation theorem of all (?1)-special systems of the form L(d, m0 , 6n ). Proof of Theorem B: In lemma 4.2 we have seen the possible cases for (?1)-curves which might split o? L(d, m0 , 6n ). Now we have to consider all these cases. To be a little bit faster we proceed along the following algorithm (see [LU02]): We go through all possible combinations of these (?1)-curves step by step. First step: If we ?nd a (?1)-curve or a (?1)-con?guration A such that L.A = ?? ≤ ?2, then we split o? the ?xed part and de?ne M = L ? ? · A. Second step: Let M′ be the residual system of M obtained by splitting o? all possible (?1)-curves. By the de?nition of (?1)-special systems we have to verify that v(M′ ) ≥ 0. We notice that the systems M are quasi-homogeneous of multiplicity ≤ 4 by lemma 4.2. Therefore we can use the results of [CM98] and [S99]. ? L = M + ? · A, v(M) ≥ 0 and M.A = 0 (1) A = L(2, 0, 15 ) and L = L(d, m0 , 65 ) This gives M = L(d ? 2n, m0 , (6 ? ?)5 ) and M.A = 0 gives d =
30?? 2 .

If ? = 2 =? d = 14 and we get m0 = 0 and v(M) = 15 with M = L(10, 0, 45 ), non-special by [S99] m0 = 1 and v(M) = 14 with M = L(10, 1, 45 ), ′′ m0 = 2 and v(M) = 12 with M = L(10, 2, 45 ), ′′ m0 = 3 and v(M) = 9 with M = L(10, 3, 45 ), ′′ m0 = 4 and v(M) = 5 with M = L(10, 4, 45 ), ′′ m0 = 5 and v(M) = 0 with M = L(10, 5, 45 ), ′′ ? = 3 is not possible because of M.A = 0. If ? = 4 =? d = 13 and we conclude m0 = 0 and v(M) = 5 with M = L(7, 0, 25 ), non-special by [CM98] m0 = 1 and v(M) = 4 with M = L(7, 1, 25 ) , ′′ m0 = 2 and v(M) = 2 with M = L(7, 2, 25 ) , ′′ m0 = 3 and v(M) = ?1

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? = 5 is not possible because of M.A = 0. From ? = 6 =? d = 12 and m0 = 0, v(M) = 0 for M = L(0, 0). (2) A = L(e, e ? 1, 12e ) e ≥ 1 and L = L(d, m0 , 62e ) Then follows M = L(d ? ? · e, m0 ? ? · e + ?, (6 ? ?)2e ) and M.A = 0 gives ?e · m0 + e · d ? 12e + m0 + ? = 0 =? m0 > d ? 12. v(M) ≥ 0 gives d ≥ m0 + ? ? 2. ? = 2 =? d ≥ m0 > d ? 12 m0 v(M) ≤ ?1 and M non-special ′′ d ′′ d?1 m0 from M.A = 0 ? d residual system d?2 10e M = L(8e, 8e, 42e ) irregular by [S99] ? non-empty d?3 9e + 1 M = L(7e + 1, 7e, 42e ) irregular by [S99] ? non-empty d?4 8e + 2 M = L(6e + 2, 6e, 42e ) irregular by [S99] ? non-empty d?5 7e + 3 M = L(5e + 3, 5e, 42e ) regular by [S99] and v(M) = 9 d?6 6e + 4 M = L(4e + 4, 4e, 42e ) regular by [S99] and v(M) = 14 d?7 5e + 5 M = L(3e + 5, 3e, 42e ) regular by [S99] and v(M) = ?2e + 20 d?8 4e + 6 M = L(2e + 6, 2e, 42e ) regular by [S99] and v(M) = ?6e + 27 d?9 3e + 7 M = L(e+7, e, 42e ) regular by [S99] and v(M) = ?12e + 35 d ? 10 2e + 8 M = L(8, 0, 42e ) regular by [S99] and v(M) = ?20e + 44 d ? 11 e+9 ? m0 ≤ ?1 not possible ? = 3 =? d ? 1 ≥ m0 > d ? 12 m0 v(M) ≤ ?1 and M non-special ′′ d?1 ′′ d?2 m0 from M.A = 0 ? d residual system d?3 9e M = L(6e, 6e, 32e ) irregular [CM98] and e(M) = 0 d?4 8e + 1 M = L(5e + 1, 5e, 32e ) irregular [CM98] and e(M) = 2 d?5 7e + 2 M = L(4e + 2, 4e, 32e ) regular [CM98] and v(M) = 5 d?6 6e + 3 M = L(3e + 3, 3e, 32e ) regular [CM98] and v(M) = 9

by by by by

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from M.A = 0 ? d residual system 5e + 4 M = L(2e + 4, 2e, 32e ) regular by [CM98] and v(M) = ?2e + 14 d?8 4e + 5 M = L(e + 5, e, 32e ) regular by [CM98] and v(M) = ?6e + 20 d?9 3e + 6 M = L(6, 0, 32e ) regular by [CM98] and v(M) = ?12e + 27 d ? 10 2e + 7 ? m0 ≤ ?1 not possible m0 d?7 ? = 4 =? d ? 2 ≥ m0 > d ? 12 m0 from M.A = 0 ? d residual system d?2 10e ? 2 M = L(6e ? 2, 6e, 22e ) empty d?3 9e ? 1 M = L(5e ? 1, 5e, 22e ) empty d?4 8e M = L(4e, 4e, 22e ) irregular by [CM98] and e(M) = 0 d?5 7e + 1 M = L(3e + 1, 3e, 22e ) regular by [CM98] and v(M) = 2 d?6 6e + 2 M = L(2e + 2, 2e, 22e ) regular by [CM98] and v(M) = 5 d?7 5e + 3 M = L(e + 3, e, 22e ) regular by [CM98] and v(M) = ?2e + 9 d?8 4e + 4 M = L(4, 0, 22e ) regular by [CM98] and v(M) = ?6e + 14 d?9 3e + 5 ? m0 ≤ ?1 not possible ? = 5 =? d ? 3 ≥ m0 > d ? 12 m0 from M.A = 0 ? d residual system d?3 9e ? 2 M = L(4e ? 2, 4e, 12e ) empty d?4 8e ? 1 M = L(3e ? 1, 3e, 12e ) empty d?5 7e M = L(2e, 2e, 12e ) regular by [CM98] and v(M) = 0 d?6 6e + 1 M = L(e + 1, e, 12e ) regular by [CM98] and v(M) = 2 d?7 5e + 2 M = L(2, 0, 12e ) regular by [CM98] and v(M) = ?2e + 5 d?8 4e + 3 ? m0 ≤ ?1 not possible For ? = 6 we have that d ? 4 ≥ m0 > d ? 12. Let m0 = d ? x. From M.A = 0 ? d = (12?x)e+(x?6). We notice that M = L((6?x)e+(x?6), (6?x)e, 0), which is regular. Taking into account that v(M) ≤ ?1 for all x ≤ 5 and m0 ≤ ?1 for all x ≥ 7 we get the only case: m0 = d ? 6 and M.A = 0 ? d = 6e and M = L(0, 0) is regular with v(M) = 0. (3) A = L(e, e, 1e ) and L = L(d, m0 , 6e ) This leads to M = L(d??e, m0 ??e, (6??)e ). M.A = 0 gives m0 = d+??6. If ? = 2 then we get m0 = d? 4, L = L(d, d? 4, 6e ) and M = L(d? 2e, d? 4? 2e, 4e ). From v(M) = ?20e+5d?6 =? v(M) ≥ 0 if d ≥ 6+20e . Furthermore 5 M is irregular by [S99] and of higher dimension if (1) e = 2f and d = 8f (2) e = 2f and d = 8f + 1 (3) e = 2f and d = 8f + 2.

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If ? = 3 then we get m0 = d ? 3, L = L(d, d ? 3, 6e ) and M = L(d ? 3e, d ? 3 ? 3e, 3e ). From v(M) = ?18e + 4d ? 3 =? v(M) ≥ 0 if d ≥ 3+18e . Further 4 M is irregular by [CM98] and of higher dimension if (1) e = 2f , d = 9f and e(M) = 0 or (2) e = 2f , d = 9f + 1 and e(M) = 2. If ? = 4 then m0 = d?2, L = L(d, d?2, 6e ) and M = L(d?4e, d?2?4e, 2e ). From v(M) = ?15e + 3d ? 1 =? v(M) ≥ 0 if d ≥ 1+15e . Further M 3 is irregular by [CM98] and of higher dimension if e = 2f , d = 10f and e(M) = 0. If ? = 5 then m0 = d?1, L = L(d, d?1, 6e ) and M = L(d?5e, d?1?5e, 1e ). From v(M) = ?11e + 2d =? v(M) ≥ 0 if d ≥ 11e . M is always regular by 2 [CM98]. If ? = 6 then m0 = d, L = L(d, d, 6e ) and M = L(d ? 6e, d ? 6e). v(M) = ?6e + d =? v(M) ≥ 0 if d ≥ 6e. M is always regular. The following two cases are easier to compute because we have no further parameters in the (-1)-curves. (4) A = L(6, 3, 27 ) and L = L(d, m0 , 67 ) This leads to M = L(d ? 6?, m0 ? 3?, (6 ? 2?)7 ). M.A = 0 gives m0 = 6d+??84 . Therefore ? = 3 is the only possible case: M = L(d ? 18, 2d ? 36). 3 To get v(M) ≥ 0 we need d = 18. =? L = L(18, 9, 67 ). (5) A = L(3, 0, 23 ) and L = L(d, m0 , 63 ) This leads to M = L(d ? 3?, m0 , (6 ? 2?)3 ). M.A = 0 gives d = 12 ? ?. ? = 2 We get L = L(10, m0 , 23 ) and M = L(4, m0 , 23 ). From v(M) ≥ 0 we get m0 ∈ {0, 1, 2}. All M are regular by [CM98]. ? = 3 We get L = L(9, m0 ) and M = L(0, m0 ). =? m0 = 0 and v(M) = 0. (6) A = L(12, 8, 39 ), L = L(d, m0 , 69 ) and ? = 2 This lead to =? M = L(d ? 24, m0 ? 16), which is regular. From M.A = 0 we get m0 = 3d?40 . Therefore v(M) ≥ 0 gives d ∈ {24, 25}, but only d = 24 2 and m0 = 16 is possible. L = M + 2 · A1 + 2 · A2 , v(M) ≥ 0, M non-special and M.A = 0 (1) A = L(δ, ?0 , 1n ) and A1 .A2 = 0 This leads to A1 = L(e, e ? 1, 12e ) and A2 = L(2e, 2e, 12e ). Further we have L = L(d, m0 , 62e ) and M = L(d ? 6e, m0 ? 6e + 2, 22e ). From M.A1 = 0 and M.A2 = 0 we get m0 = d ? 4 and d = 8e + 2. Therefore we have M = L(2e + 2, 2e, 22e ), which is regular by [CM98] and v(M) = 5. (2) A1 = L(δ1 , ?01 , 1n ) and A2 = L(δ2 , ?02 , 2n ) A1 .A2 = 0 gives only the following possibilities: (1) A1 ∈ L(2, 1, 14 ) and A2 ∈ L(3, 0, 23 ), (2) A1 ∈ L(2, 0, 15 ) and A2 ∈ L(3, 0, 23 ) or (3) A1 ∈ L(2, 2, 12 ) and A2 ∈ L(3, 0, 23 ).

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(1) and (2) are not possible cases as these curves are elements of quasihomogeneous systems based on a di?erent number of points with equal multiplicities. It is not possible to ?nd a suitable system L(d, m0 , 6n ). So let us focus on (3), where we see that it is equivalent to assume A2 ∈ L(3, 2, 22 ). In this case we see that L = L(d, m0 , 62 ) and M = L(d ? 10, m0 ? 8). From M.A1 = 0 and M.A2 = 0 we conclude that d = 10 and m0 = 8, that means we get the system L = L(10, 8, 62 ).

L = M + 2 · A1 + 3 · A2 , v(M) ≥ 0 and M.A = 0 A = L(δ, ?0 , 1n ) and A1 .A2 = 0 (1) A1 = L(e, e ? 1, 12e ) & A2 = L(2e, 2e, 12e ) Moreover we have L = L(d, m0 , 62e ) and M = L(d ? 8e, m0 ? 8e + 2, 12e ). From M.A1 = 0 and M.A2 = 0 we get m0 = d ? 3 and d = 9e + 1. Therefore we have M = L(e + 1, e, 12e ) which is regular by [CM98] and v(M) = 2. (2) A1 = L(2e, 2e, 12e ) & A2 = L(e, e ? 1, 12e ) Furthermore we have L = L(d, m0 , 62e ) and M = L(d ? 7e, m0 ? 7e + 3, 12e ). From M.A1 = 0 and M.A2 = 0 we get m0 = d ? 4 and d = 8e + 1. Therefore we have M = L(e + 1, e, 12e ) which is regular by [CM98] and v(M) = 2.

L = M + 2 · A1 + 4 · A2 , v(M) ≥ 0 and M.A = 0 A = L(δ, ?0 , 1n ) and A1 .A2 = 0

(1) A1 = L(e, e ? 1, 12e ) and A2 = L(2e, 2e, 12e ) Moreover we have L = L(d, m0 , 62e ) and M = L(d ? 10e, m0 ? 10e + 2). From M.A1 = 0 and M.A2 = 0 we get m0 = d ? 2 and d = 10e. Therefore we get M = L(0, 0) and v(M) = 0. (2) A1 = L(2e, 2e, 12e ) and A2 = L(e, e ? 1, 12e ) Furthermore we have L = L(d, m0 , 62e ) and M = L(d ? 8e, m0 ? 8e + 4). From M.A1 = 0 and M.A2 = 0 we get m0 = d ? 4 and d = 8e. Therefore we have M = L(0, 0) and v(M) = 0.

L = M + 2 · A1 + 2 · A2 + 2 · A3 , v(M) ≥ 0 and M.A = 0 As A1 = L(e, e ? 1, 12e ) and A2 = L(e, e, 1e ) are the only compound (?1)-con?gurations with multiplicity m = 1 in p1 , . . . , pn which have intersection multiplicity = 0. Therefore we are immediately in case 4.

This ?nally completes our proof of the classi?cation theorem.

QUASI-HOMOGENEOUS LINEAR SYSTEMS ON P WITH BASE POINTS OF MULTIPLICITY 6

11

5. The Degeneration Method In this section we give a rough overview of the degeneration of the plane as introduced by Ciliberto and Miranda in [CM98]. We refer to this paper for further details. As in every degeneration method the aim is to specialize the base points of a system L(d, m0 , mn ) in such a way that on the one hand the dimension is easier to compute but on the other hand it does not change. At ?rst we consider the geometric situation. Let ? be a complex disc around the origin. We de?ne V = P2 × ?. Let p1 : V ?→ P2 and p2 : V ?→ ? be the projections. Now we blow up a line L in V0 = p?1 (0) (f : X ?→ V ) and obtain the following situation with 2 πi = f ? p i : X0 = P ∪R F X: ::  :  f ::: :: π2   π1  V LL ::: LL  ~~ LL :::  ~~~ LL :  ~ p1 p2 LLL:: ×~~~ L7 ( ? F? ?
?? σ ?? ?? 1

P2

P2

?1 ?1 ? Now Xt = π2 (t) = P2 for all t = 0. X0 = π2 (0) is a union of two surfaces, the strict transform of V0 ? P2 (called P) and the exceptional divisor F = f ?1 (L). F is isomorphic = to the blow-up of P2 in one point p (here via σ). The surfaces are glued together along the line R, which can be identi?ed with L in P and with the exceptional divisor E = σ ?1 (p) in F. ? As in [CM98] we de?ne OX (d) = π1 OP2 (d) and OX (d, k) = OX (d) ?OX OX (kP). We set χ(d, k) = OX (d, k)|X0 . Let H be the pull-back of a general line in P2 via σ. Then we have OX (d, k)|Xt ? OP2 (d) for t = 0. Furthermore χ(d, k)|P ? OP2 (d ? k) and χ(d, k)|F ? = = = OF (dH ? (d ? k)E).

We ?x n ? b + 1 general points p0 , p1 , . . . , pn?b on P and b general points pn?b+1 , . . . , pn on F. We de?ne L0 to be the linear sub-system of χ(d, k) de?ned by all divisors of χ(d, k) having multiplicity at least m0 at p0 and at least m at the points p1 , . . . , pn (write L0 = L(d, m0 , mn?b , mb )). We say that L0 is obtained from L = L(d, m0 , mn ) by an (k,b)-degeneration. L0 can be considered as a ?at limit on X0 of L. By semi-continuity we obtain ?0 = ?(L0 ) ≥ ?(L). In particular, if ?0 = e(L) then L is non-special. Now L0 restricts on P to a system LP = L(d ? k, m0 , mn?b ). Furthermore we restrict L0 on F to LF = L(d, d ? k, mb ) (the identi?cation we obtain by blowing down LF to P2 via σ). Now we de?ne as in [CM98] RP to be the linear system on R obtained by restricting LP to R. We have the following exact sequence
|L ? +L 0 ?→ LP ?→ LP ?→ RP ?→ 0.

? The kernel system LP consists of all divisors having L as component. So we can identify ?P = L(d ? k ? 1, m0 , mn?b ). L ? We analogously de?ne RF and obtain LF = L(d, d ? k + 1, mb ) (parametrising the divisors in LF which have E as a component). Let us recall some further abbreviations from [CM98]: 5.1 De?nitions

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MICHAEL KUNTE

vP = v(LP ), vF = v(LF ), ? ? vP = v(LP ), vF = v(LF ), ? ? ?P = ?(LP ), ?F = ?(LF ), ? ? ? ? ?P = ?(LP ), ?F = ?(LF ), ? rP = ?P ? ?P ? 1, the dimension of RP , ?F ? 1, the dimension of RF . rF = ?F ? ? In [CM98] it is shown that the associated vector spaces to RP and RF are transversal subspaces of Γ(R, OR (d ? k)). This leads to the following corollary: 5.2 Corollary (Key-Lemma on ?0 ) We have two cases: ? ? (1) If rP + rF ≤ d ? k ? 1, then ?0 = ?P + ?F + 1. (2) If rP + rF ≥ d ? k ? 1, then ?0 = ?P + ?F ? d + k. A proof can be found in [CM98] 6. Proof of the Main Theorem Before giving the proof let us state two lemmas which are corollaries of the Key-Lemma 5.2. The proof of these is given for an analogous case in [LU02]. 6.1 Lemma (case v(L) ≤ ?1) Let L = L(d, m0 , 6n ) with v(L) ≤ ?1. If there are integers k (k < d) and b (b < n) such that a (k, b)-degeneration can be found with the following properties of the restrictions of L0 ? LF and LP are both non-special, and ? ? ? the kernel systems LF and LP are empty with vP ≤ v(L), ? then L is empty. 6.2 Lemma (case v(L) ≥ ?1) Let L = L(d, m0 , 6n ) with v(L) ≥ ?1. If there are integers k (k < d) and b (b < n) such that a (k, b)-degeneration can be found with ? LF and LP are both non-special, vP ≥ ?1, vF ≥ ?1, and ? ? ? ? ? the kernel systems LF and LP have the property v(L) ? 1 ≥ ?P + ?F , then L is non-special. The following three lemmas state parts of the result of the Main Theorem A. We prove them independently later on. 6.3 Lemma (three base points) A linear system L(d, m0 , mn ) with at most three base points (n ≤ 2) is special if and only if it is (?1)-special. 6.4 Lemma (large multiplicities m0 in p0 ) Let d ≥ 25. If m0 ≥ d ? 9 then L(d, m0 , 6n ) is special if and only if it is (?1)-special. 6.5 Lemma (low degrees) If d ≤ 140 then L(d, m0 , 6n ) is special if and only if it is (?1)-special.

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13

Proof of the Main Theorem A: Let L = L(d, m0 , 6n ). By the lemma for large multiplicities (6.4) we can assume that d ≥ m0 + 10 ≥ 10. Furthermore by the lemma for low degrees (6.5) the statement is true for d ≤ 140. We can assume d ≥ 141. We continue by induction on d where 6.5 can be considered as the base of the induction. As all such L are not (?1)-special we have to show that L is non-special. The method is to get the system L0 on the special ?ber by a degeneration of L. With Lemmas 6.1 and 6.2 we can prove the regularity of L if the restrictions of L0 to P and to F have certain properties. These properties can be achieved as the main conjecture holds for the systems on P by induction and for the ones on F by 6.4. We perform now a (5, b)-degeneration on L and get the following systems on the special ?ber: P: LP = L(d ? 5, m0 , 6n?b ) F: LF = L(d, d ? 5, 6b ) ? LP = L(d ? 6, m0 , 6n?b ) Step 1 (case v(L) ≤ ?1): We want to apply Lemma 6.1 for the case v(L) ≤ ?1. ? First of all we need to have LF empty. By the lemma for large multiplicities in m0 (6.4) ? we have that LF is non-special if it is non-(?1)-special. Therefore by our classi?cation theorem B it is su?cient to choose d < 4b, i.e., b > d . Also we get vF ≤ ?1, which means ? 4 this system is empty. Next let us ?nd a su?cient condition to get vP ≤ v(L). A computation gives vP ? v(L) = ? ? ?6d + 21b + 9, hence it is su?cient to have ?6d + 21b + 9 ≤ 0, that is b ≤ 6d?9 . 21 Now we want to ?nd su?cient conditions to have LF non-special. By 6.4 this is already the case if we ?nd conditions for LF not to be (?1)-special. By Theorem B it is su?cient 2 to force d > 7b + 3, that is b < 7 (d ? 3). As 2 (d ? 3) ≤ 6d?9 , this new condition on b 2 7 21 includes also vP ≤ v(L). ? In the next step we are searching for a su?cient condition to get LP non-special. By induction on d LP = L(d ? 5, m0 , 6n?b ) is special if and only if it is (?1)-special. By our list in Theorem B we notice that LP is non-(?1)-special if we choose n ? b odd as we have assumed that d ? m0 ≥ 10 and d ≥ 141. ? In the last step we look for a su?cient condition on b to get LP empty. Here we have to be more careful. When d ? m0 ≥ 11 we get for the same reasons as in the case of LP that ? LP is non-special if n ? b is odd. When d ? m0 = 10 then from Theorem B we know that ?P = ?20(n ? b) + 5(d ? 6) ? 6 if n ? b is odd. That means we want this expression to ? 1 ? be negative. From ?P ≤ ?1 ?? b ≤ 4 (7 ? d) + n we get a su?cient condition on b. As by assumption v(L) ≤ ?1, we can conclude that v(L) = 11d ? 21n ? 45 ≤ ?1. Therefore n ≥ 11d?44 . That means we can formulate the above condition on b without n (using a 21 1 29 lower bound on n) and get b ≤ 4 (7 ? d) + 11d?44 = ? 84 + 23d . 21 84 Let us now reformulate all su?cient conditions (separated for the cases d ? m0 = 10 and d ? m0 > 10) in a compact form: If d ? m0 > 10 we ?nd a b such that we can apply Lemma 6.1 if 6 1 2 d ? ? d > 2 ?? d ≥ 81. 7 7 4 ? LF = L(d, d ? 4, 6b )

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MICHAEL KUNTE

If d ? m0 = 10 we ?nd also a b to apply 6.1 if ? 29 23 1 + d ? d > 2 ?? d ≥ 99. 84 84 4

Step 2 (case v(L) ≥ ?1): We want to use Lemma 6.2 for the case v(L) ≥ ?1. Still all notations are with respect to the above (5, b)-degeneration. In a ?rst step we want to ?nd a su?cient condition on b to get LP non-special. Exactly as in step 1 we get by induction that LP is non-special if we choose b such that n ? b is odd, because we assume d ? m0 ≥ 10 and d ≥ 141. Next we want to ?nd su?cient conditions on b to get the system LF non-special and vF ≥ ?1. By the lemma for large multiplicities (6.4) in m0 we have again as above that LF is non-special if and only if it is non-(?1)-special. We conclude that we get LF non6 special if we have d > 7 b + 3, that is if b < 2 d ? 7 , by Theorem B. As vF = 6d ? 21b ? 10 2 7 2 we see that vF ≥ ?1 which is equivalent to b ≤ 7 d ? 3 . Therefore the condition for getting 7 LF non-special gives already that vF ≥ ?1. From Theorem B we note again that b >
d 4

? con?rms that LF is non-special and vF ≤ ?1. ?

? ? Let us now consider LP : As above LP is by induction non-special if n ? b is odd and d ? m0 ≥ 11. In the case d ? m0 ≥ 11 we force also vP ≤ v(L), that is b ≤ 6d?9 . In the ? 21 ? case d ? m0 = 10 we conclude - exactly as above - that if n ? b is odd LP is non-special ? or ?P = ?20(n ? b) + 5(d ? 6) ? 6. Therefore we force ?20(n ? b) + 5(d ? 6) ? 6 ≤ ?1, that means b ≤ 1 (7 ? d) + n. As we are in the case v(L) ≥ ?1 we have the equation 4 11d ? 21n ? 45 ≥ ?1 which means n ≤ 11d?44 . It is enough to check the independence of 21 all conditions on the base points in L for the highest possible number n of points. We ?x ? this n and use a lower bound 11d?44 ? 1 of it. That means a su?cient condition for LP to 21 1 11d?44 ?113+23d be non-special is b ≤ 4 (7 ? d) + 21 ? 1 = . 84 To ful?ll all these conditions we need to have d large enough. All together this gives so far: If d ? m0 ≥ 11 we are able to ?nd a su?cient b if 2 6 1 d ? ? d > 2 ?? d ≥ 81. 7 7 4 If d ? m0 = 10 we are able to ?nd a su?cient b if 113 1 23 d? ? d > 2 ?? d ≥ 141. 84 84 4 In both cases we have that LP and LF are non-special and vF ≥ ?1. From vF ≤ ?1 and ? ?F and LP are ? from vP = v ? 1 ? vF we get immediately vP ≥ ?1. We have v ≥ vP . As L ? ? non-special we are able to conclude the following two cases: If vP ≤ ?1 then ? ? ? ?P + ?F = ?2 ≤ v ? 1, and if vP ≥ ?1 then ? ? ? ? ?P + ?F = vP + ?F ≤ v ? 1. ? In both cases we are able to apply Lemma 6.2 and conclude that L is non-special.

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15

7. Proof of the Lemmas Before starting the proofs we should take some time to explain the use of Quadratic Cremona Transformations for our purpose. We identify such a transformation with blowing up three general points and blowing down their connecting lines. Such a transformation is called to be based on the three points. Furthermore one can see by the blow-up and -down interpretation that a linear system L(d, m0 , m1 , m2 , m3 , . . . , mn ) is transformed by a Cremona transformation based on the points p0 , p1 , p2 to a system L(2d ? m0 ? m1 ? m2 , d ? m1 ? m2 , d ? m0 ? m2 , d ? m0 ? m1 , m3 , . . . , mn ). If all involved numbers are non-negative (see [CM98]), the dimension and the virtual dimension of a system L do not change under Cremona transformations. In fact a (?1)-curve splitting o? a system L is transformed again into a (?1)-curve, which splits o? the transformed system. Therefore it is equivalent to examine a system L or its Cremona transformed for our purpose. We use suitable sequences of Cremona transformations in the following proofs to obtain systems which are already examined in previous papers. Proof of the lemma of three base points 6.3: This can be seen by direct computations with base points (1 : 0 : 0), (0 : 1 : 0) and (0 : 0 : 1). Of course, the statement is also included in the result in [H89]. Proof of the lemma of large multiplicities m0 in p0 6.4: We consider the system L(d, m0 , 6n ). For the case of m0 ≥ d ? 7 [CM98, Proposition 6.2., Corollary 6.3., Proposition 6.4.] give a classi?cation of the special systems of this type. Comparing it with our list in Theorem B gives the statement. Now let d ≥ 25. The strategy for the proof is to perform a sequence of Cremona transformations in order to get systems, which can be examined easier. Furthermore we apply the degeneration method again and use again Cremona transformations to prove regularity of some of the obtained systems. case: d ? m0 = 8 Let L = L(d, d ? 8, 6n ). We note that if we perform k Cremona transformations, based on p0 and successively on two other base points of multiplicity 6, we obtain that it is now equivalent to consider the Cremona transformed system (for the strategy see [LU02]): L ? L(d ? 4k, d ? 8 ? 4k, 6n?2k , 22k ) We set d ? 8 = 4t + ? with ? ∈ {0, 1, 2, 3}. And n = 2q + η with η ∈ {0, 1}. If t ≤ q we perform k = t transformations on L(d, d ? 8, 6n ) based on p0 and successively two other base points of multiplicity 6 and obtain L ? L(8 + ?, ?, 6n?2t , 22t ). The system on the right hand side is of bounded multiplicity, that means all multiplicities are ≤ 6. Such systems are special if and only if they are (?1)-special by [Y03]. If t > q we perform k = q transformations on L(d, d ? 8, 6n ) again based on p0 and successively two other base points of multiplicity 6 and obtain L ? L(d ? 4q, d ? 8 ? 4q, 6η , 22q ). If η = 0 we are in the case of quasi-homogeneous linear systems of multiplicity 2, here the main conjecture is true by [CM98]. If η = 1 we have to examine systems of the type L = L(δ, δ ? 8, 6, 22q ) with δ = d ? 4q. Now let us perform a (2, b)-degeneration and get the following systems: LP = L(δ ? 2, δ ? 8, 6, 22q?b ) LF = L(δ, δ ? 2, 2b )

16

MICHAEL KUNTE

? ? LP = L(δ ? 3, δ ? 8, 6, 22q?b ) LF = L(δ, δ ? 1, 2b ) If v(L) ≤ ?1 we want to apply lemma 6.1. By our classi?cation Theorem B there is no (?1)-special system of the type L(d, d ? 8, 6n ) if d ≥ 25. That means we have to show that the system L is empty. To use 6.1 we have again to consider all the systems obtained by the degeneration as in the proof of the main theorem. ? ? In a ?rst step let us consider LF . As LF is a quasi-homogeneous system of multiplicity m = 2 we see in [CM98], that this system is never special. Then vF = 2δ ? 3b leads to a ? ? su?cient condition to get LF empty. This condition is b ≥ 2δ+1 . 3 In a next step we want to ?nd a su?cient condition to get LF non-special. This is true by [CM98] if b is odd. So let us force b to be odd as a su?cient condition for this case. Now we consider LP . We claim: LP is non-special. To show the claim we apply at ?rst a Cremona transformation based on the points of multiplicity δ ? 8, 6 and on one point of multiplicity 2. This leads to the following system: LP ? L(δ ? 4, δ ? 10, 4, 22q?b?1 ). Above we forced b to be odd, therefore we assume 2q ? b ? 1 ≥ 2 (otherwise skip this step) is even. Now we apply successively 2q?b?1 Cremona transformations, based in p0 and two 2 points of multiplicity 2. Therefore we see that we have the following equivalence: LP ? L(δ ? 4 + 2q ? b ? 1, δ ? 10 + 2q ? b ? 1, 42q?b ). From δ = d ? 4q ≥ 12 + ? we get by [S99, Theorem 2.1, Theorem 5.2] that this system is never special. ? ? Finally we have to consider LP . Again we claim that LP is never special. We have by the above assumption that 2q ? b is odd. At ?rst we split o? the line through the points of multiplicity δ ? 8 and 6. As the virtual dimension doesn’t change we get ? LP ? L(δ ? 4, δ ? 9, 5, 22q?b ). Another Cremona transformation based in p0 , p1 and one point of multiplicity 2 leads to the equivalence ? LP ? L(δ ? 6, δ ? 11, 3, 22q?b?1 ). Now as in the case of LP we apply another 2q?b?1 Cremona transformations based in p0 2 and successively in two points of multiplicity 2. We end up with the equivalence: 2q ? b ? 1 2q ? b ? 1 2q?b ? LP ? L(δ ? 6 + , δ ? 11 + ,3 ). 2 2 Now we are able to conclude with [CM98] - as we are in the case of a quasi-homogeneous system of multiplicity 3 - that this system is never special. To apply 6.1 we have to ?nd a su?cient condition for b to get vP ≤ ?1, therefore it is ? su?cient to have vP ? v(L) ≤ 0, which is equivalent to b ≤ δ. ? All together we ?nd a su?cient b if δ ? 2δ+1 ≥ 2 ?? δ ≥ 8. As we have seen above 3 we have already δ ≥ 12 + ?. This means we can apply Lemma 6.1 and conclude that L(d, d ? 8, 6n ) is empty in the case v(L) ≤ ?1. Now we have to consider the case v(L) ≥ ?1. Here we want to apply the Lemma 6.2. As in the case v(L) ≤ ?1 we can always ?nd a b such that all the systems obtained by the above (2, b)-degeneration are non-special. Let us choose such a b like above and then ? ? consider the systems LP , LP , LF and LF . From vP = v(L)? vF ?1, vF ≤ ?1 and v(L) ≥ ?1 ? ? we conclude vP ≥ v(L) ≥ ?1. A direct computation gives vF ≥ ?1.

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17

? As the inequality vP ≤ v(L) is also ful?lled we get ?P ≤ v(L). Therefore we can apply ? n ) is non-special. Lemma 6.2 and conclude that L(d, d ? 8, 6 case: d ? m0 = 9 Let L = L(d, d ? 9, 6n ). We note as above that if we perform k Cremona transformations, based on p0 and successively on two other base points of multiplicity 6, we obtain that: L ? L(d ? 3k, d ? 9 ? 3k, 6n?2k , 32k ) We set d ? 9 = 3t + ? with ? ∈ {0, 1, 2}. And n = 2q + η with η ∈ {0, 1}. If t ≤ q we perform k = t transformations on L(d, d ? 9, 6n ) based on m0 and successively on two other base points of multiplicity 6 and obtain L ? L(9 + ?, ?, 6n?2t , 32t ). Then the system on the right hand side is of bounded multiplicity, that means all multiplicities are ≤ 6. As mentioned above such systems are special if and only if they are (?1)-special by [Y03]. If t > q we perform k = q transformations on L(d, d ? 9, 6n ) and obtain L ? L(d ? 3q, d ? 9 ? 3q, 6η , 32q ). If η = 0 we are in the case of quasi-homogeneous linear systems of multiplicity 3, here the main conjecture is true by [CM98]. If η = 1 we have to examine systems of the type L(δ, δ ?9, 6, 32q ) with δ = d?3q. If δ < 15 we are in the case of systems of bounded multiplicity where the main conjecture holds by [Y03]. So we can assume δ ≥ 15. Also we can assume q ≥ 1 (otherwise the statement is clear). Now let us perform a (3, b)-degeneration and get the following systems: LP = L(δ ? 3, δ ? 9, 6, 32q?b ) LF = L(δ, δ ? 3, 3b ) ? ? LP = L(δ ? 4, δ ? 9, 6, 32q?b ) LF = L(δ, δ ? 2, 3b ) If v(L) ≤ ?1 we again want to apply Lemma 6.1. So let as go through all the systems from the above (3, b)-degeneration and search for su?cient conditions on b to apply Lemma 6.1. ? Let us consider LF at ?rst. Here it is su?cient to choose b > ? special by [CM98] and ?F = ?1. Then we force (to apply 6.1) vP ≤ v. This is ful?lled if b ≤ ?
δ 2

to get this system non-

In a next step consider LF . By [CM98] this is non-special if b is odd.
2δ?1 3 .

Now let us consider LP . We claim that this system is never special. To see that let us perform Cremona transformation based on the points of multiplicity δ ? 9, 6 and 3. We obtain: LP ? L(δ ? 6, δ ? 12, 32q?b?1 ) These systems are always regular by [CM98] as we have δ high enough. ? A little bit more complicated is the case of LP . We are searching for a su?cient condition ?P empty. We want to show, that LP is never special. Then we get the ? on b to get L condition simply be choosing b such that vP ≤ ?1 (ful?lled by vP ≤ v). ? ? First of all we split o? a line through p0 , the point of multiplicity m0 = δ ? 9, and the point of multiplicity 6. Therefore we obtain ? LP ? L(δ ? 5, δ ? 10, 5, 32q?b ).

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MICHAEL KUNTE

as the virtual dimension doesn’t change in that case. If 2q ? b > 0 applying a further Cremona transformation based in the points of multiplicity δ ? 10, 5 and one point of multiplicity 3 gives ? LP ? L(δ ? 8, δ ? 13, 2, 32q?b?1 ). Note that 2q ? b ? 1 is an even number, as b is odd. We apply now successively Cremona transformations, based on the point p0 and on two other points of multiplicity 3. It is better again to consider two di?erent cases. At ?rst assume δ ? 13 ≥
2q?b?1 . 2

Then we get

2q ? b ? 1 2q?b 2q ? b ? 1 ? , δ ? 13 ? ,2 ). LP ? L(δ ? 8 ? 2 2 By [CM98] such a system is never special. Secondly assume δ ? 13 < we obtain:
2q?b?1 . 2

Let m = δ ? 13. Then after m such transformations

? LP ? L(5, 22m+1 , 32q?b?1?2m ). Again splitting o? a line through two points of multiplicity 3 (virtual dimension does not change) gives: ? LP ? L(4, 22(m+1)+1 , 32q?b?1?2(m+1) ). Now two if 2q ? b ? 1 ? 2(m + 1) ≥ 2 splitting o? lines gives that LP is empty. Secondly if 2q ? b ? 1 ? 2(m + 1) = 0 we have also by [CM98] that the system is empty (as m ≥ 2 by assumption that δ ≥ 15). Taking into account all our conditions on b we require 2δ ? 1 δ ? > 2 ?? δ ≥ 15. 3 2 Finally applying Lemma 6.1 gives that the system L = L(δ, δ ? 9, 6, 32q ) is empty in the case v ≤ ?1. Now we have to consider the case v(L) ≥ ?1. We want to apply Lemma 6.2. ? ? As in the case of v ≤ ?1 we get that all the systems LF , LF , LP and LP are non-special ?F = ?1 and vP ≤ v with a suitable b for the degeneration. and ? ? ? That means here ?P ≤ v and we can apply Lemma 6.2 and conclude that v(L) = ?, that means L = L(δ, δ ? 9, 6, 32q ) is regular. This ?nally completes our proof for the case of multiplicities m0 = d ? 8 and m0 = d ? 9.

Proof of the lemma of low degrees 6.5: The main tool for this proof is a computer program which uses (5, b)- and (6, b)degenerations of the plane in order to prove that certain non-(?1)-special systems are non-special. This algorithm is given by Laface and Ugaglia in [LU02]. We implemented this algorithm in Singular (see [Sing]). Furthermore to treat the cases where the degenerationmethod fails we implemented a method used by Yang in [Y03]. This method specializes the base points on a line and moves them to in?nity. Then it is easier to check if the given conditions on the base points are independent. If this is still the case it proves regularity of a given system. Below we list only the cases in which the program fails. All these but 10 cases are solved by ad-hoc methods (mainly Cremona transformations). The remaining 10 cases we computed directly with Singular in characteristic 32003. One can see that this implies then regularity in characteristic 0, too.

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19

d ? m0 8 8 14 13 12 11 10 8 15 15 14 13 12 11 10 9 9 8 16 16 15 14 13 12 11 10 10 9 8 17 16 15 11 10 9 8 19 18 17 15 14 13 12 9 8 12

system
L = L(8, 0, 63 ) L = L(9, 1, 63 ) L = L(14, 0, 66 ) L = L(14, 1, 66 ) L = L(14, 2, 66 ) L = L(14, 3, 66 ) L = L(14, 4, 66 ) L = L(14, 6, 65 ) L = L(15, 0, 67 ) L = L(15, 0, 66 ) L = L(15, 1, 66 ) L = L(15, 2, 66 ) L = L(15, 3, 66 ) L = L(15, 4, 66 ) L = L(15, 5, 66 ) L = L(15, 6, 66 ) L = L(15, 6, 65 ) L = L(15, 7, 65 ) L = L(16, 0, 68 ) L = L(16, 0, 67 ) L = L(16, 1, 67 ) L = L(16, 2, 67 ) L = L(16, 3, 67 ) L = L(16, 4, 67 ) L = L(16, 5, 67 ) L = L(16, 6, 67 ) L = L(16, 6, 66 ) L = L(16, 7, 66 ) L = L(16, 8, 66 ) L = L(17, 0, 68 ) L = L(17, 1, 68 ) L = L(17, 2, 68 ) L = L(17, 6, 67 ) L = L(17, 7, 67 ) L = L(17, 8, 67 ) L = L(18, 10, 67) L = L(19, 0, 610 ) L = L(19, 1, 610 ) L = L(19, 2, 610 ) L = L(19, 4, 69 ) L = L(19, 5, 69 ) L = L(19, 6, 69 ) L = L(19, 7, 69 ) L = L(19, 10, 67) L = L(19, 11, 67) L = L(20, 8, 69 )

dimension
?1 ?1 ?1 ?1 ?1 ?1 ?1 ?1 ?1 > ?1 > ?1 > ?1 > ?1 ?1 ?1 ?1 > ?1 > ?1 ?1 > ?1 > ?1 > ?1 ?1 ?1 ?1 ?1 > ?1 ?1 ?1 > ?1 > ?1 ?1 > ?1 ?1 ?1 ?1 ?1 ?1 ?1 > ?1 > ?1 ?1 ?1 > ?1 ?1 > ?1

method 3-point lemma splitting o? lines Cremona and splitting o? as L(14, 0, 66) is empty as L(14, 0, 66) is empty as L(14, 0, 66) is empty as L(14, 0, 66) is empty as L(14, 0, 66) is empty Cremona as L(15, 3, 66) is regular as L(15, 3, 66) is regular as L(15, 3, 66) is regular Cremona and [CM98] Cremona and splitting o? as L(15, 4, 66) is empty as L(15, 4, 66) is empty as L(15, 0, 66) is regular Cremona and [CM98] as L(16, 3, 67) is empty as L(16, 2, 67) is regular as L(16, 2, 67) is regular Cremona and [CM98] Cremona and splitting o? as L(16, 3, 67) is empty as L(16, 3, 67) is empty as L(16, 3, 67) is empty as L(16, 2, 67) is regular Cremona and splitting o? as L(16, 7, 66) is empty as L(17, 1, 68) is regular Cremona Cremona and splitting o? as L(17, 1, 68) is regular Cremona and splitting o? as L(17, 7, 67) is empty Cremona and splitting o?
[CM00] as L(19, 0, 610) is empty as L(19, 0, 610) is empty as L(19, 5, 69) is regular regular by [Y03] as L(19, 0, 610) is empty as L(19, 0, 610) is empty Cremona and [CM00]

lines

lines

lines

lines

lines lines lines

Cremona and splitting o? lines direct computation with [Sing] in char= 32003

20

MICHAEL KUNTE

d ? m0 11 8 11 10 9 8 22 21 20 19 16 15 13 11 10 10 9 8 12 10 9 8 10 9 8 13 10 12 10 13 10 10 13 10 10

system
L = L(20, 9, 69) L = L(20, 12, 67) L = L(21, 10, 69) L = L(21, 11, 69) L = L(21, 12, 68) L = L(21, 13, 68) L = L(22, 0, 613) L = L(22, 1, 613) L = L(22, 2, 613) L = L(22, 3, 613) L = L(22, 6, 612) L = L(22, 7, 612) L = L(22, 9, 611) L = L(22, 11, 610) L = L(22, 12, 610) L = L(22, 12, 69) L = L(22, 13, 69) L = L(22, 14, 69) L = L(23, 11, 611) L = L(23, 13, 610) L = L(23, 14, 69) L = L(23, 15, 69) L = L(24, 14, 610) L = L(24, 15, 610) L = L(24, 16, 610) L = L(25, 12, 613) L = L(25, 15, 611) L = L(26, 14, 613) L = L(29, 19, 613) L = L(31, 18, 617) L = L(31, 21, 614) L = L(38, 28, 618) L = L(40, 27, 623) L = L(40, 30, 619) L = L(46, 36, 622)

dimension
?1 > ?1 > ?1 ?1 > ?1 ?1 > ?1 > ?1 ?1 ?1 > ?1 ?1 ?1 ?1 ?1 > ?1 ?1 ?1 > ?1 ?1 > ?1 ?1 > ?1 ?1 ?1 ?1 ?1 ?1 > ?1 ?1 > ?1 ?1 ?1 ?1 ?1

method Cremona and [LU02] Cremona and [CM00] Cremona and [Y03] Cremona and [Y03] Cremona and [CM00] Cremona, splitting o? lines and [CM00] as L(22, 1, 613 ) is regular
[Y03] [Y03] as L(22, 2, 613 ) is empty as L(22, 1, 613 ) is regular

direct computation with [Sing] in char = 32003
′′

Cremona and [Y03] as L(22, 11, 610) is empty Cremona and [S99] Cremona and splitting o? lines as L(22, 13, 69) is empty direct computation with [Sing] in Cremona and [S99] Cremona and [CM00] Cremona and splitting o? lines Cremona and [CM00] Cremona and [Y03] as L(24, 15, 610) is empty direct computation with [Sing] in Cremona and [Y03] direct computation with [Sing] in direct computation with [Sing] in
′′

char = 32003

char = 32003 char = 32003 char = 32003

Cremona and [S99] Cremona and [S99] direct computation with [Sing] in char = 32003
′′

Cremona and [S99]

References
Ciliberto, C.: Geometric Aspects of Polynomial Interpolation in More Variables and of Waring’s Problem. Proceedings of the ECM, Barcelona (2000). [CM98] Ciliberto, C.; Miranda, R.: Degenerations of Planar Linear Systems. J. Reine Angew. Math. 501, 191-200 (1998). [CM00] Ciliberto, C.; Miranda, R.: Linear systems of plane curves with base points of equal multiplicity. Transactions of A.M.S. 352, 4037-4050 (2000). [Sing] Greuel, G.; P?ster, G.; Sch¨nemann, H.: Singular 2.3, A Computer Algebra System for Polynoo mial Computations. Zentrum f¨r Computeralgebra Technische Universit¨t Kaiserslautern (2002), u a http://www.singular.uni-kl.de. [H89] Harbourne, B.: Free resolutions of fat point ideals on P2 . J. Pure Appl. Algebra 125, no. 1-3, 213-234 (1989). [C00]

QUASI-HOMOGENEOUS LINEAR SYSTEMS ON P WITH BASE POINTS OF MULTIPLICITY 6

21

[L99] Laface, A.: Linear Systems with ?xed base points of given multiplicity. PhD Thesis Rom (1999). [LU02] Laface, A.; Ugaglia, L.: Quasi-homogeneous linear systems on P2 with base points of multiplicity 5. Preprint arXiv:math.AG 0205270 v1 (2002). [S99] Seibert, J.: The dimension of quasi-homogeneous linear systems with multiplicity four. Preprint arXiv:math.AG 9905076 v1 (1999). [Y03] Yang, S.: Linear systems of plane curves through base points of bounded multiplicity. Preprint Summer-school Torino (2003). ¨ ¨ Fachbereich Mathematik, Technische Universitat Kaiserslautern, Erwin-SchrodingerStra?e, 67663 Kaiserslautern E-mail address: kunte@mathematik.uni-kl.de


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