PROJECTIVE BUNDLES OF SINGULAR PLANE CUBICS STEFAN KEBEKUS

PROJECTIVE BUNDLES OF SINGULAR PLANE CUBICS STEFAN KEBEKUS ABSTRACT. Classification theory and the study of projective varieties which are covered by rational curves of minimal degrees naturally leads to the study of families of singular rational curves. Since families of arbitrarily singular curves are hard to handle, it has been shown in [Keb00] that there exists a partial resolution of singularities which transforms a bundle of possibly badly singular curves into a bundle of nodal and cuspidal plane cubics. In cases which are of interest for classification theory, the total spaces of these bundles will clearly be projective. It is, however, generally false that an arbitrary bundle of plane cubics is globally projective. For that reason the question of projectivity and the study of moduli seems to be of interest, and the present work gives a characterization of the projective bundles. CONTENTS 1. Introduction 1 2. Ruled surfaces and transformations of ruled surfaces 2 3. Characterization of projective bundles 4 4. Projective bundles and osculating points 6 5. Proof of theorem 3.1 9 6. Proof of theorem 3.3 9 7. Proof of theorem 3.4 9 References 11 1. INTRODUCTION Let C be a smooth algebraic curve and π : X C be a morphism from a (singular) algebraic variety to C. We assume that every fiber of π is isomorphic to an irreducible and reduced singular plane cubic. Omitting the word integral for brevity, we call this setup a bundle of singular plane cubics. Although C can always be covered by open subsets U α such that X α := π 1 (U α ) can be identified with a family of cubic curves in P 2 U α, it is generally not true that X can be embedded into a P 2 -bundle over C. In fact, X even need not be projective. The aim of this paper is to characterize those bundles which are projective. It turns out that the set P of projective bundles is neither open nor closed in the set of all bundles of singular plane cubics and that the irreducible components of P are subvarieties of high codimension. For a precise statement, see remark 3.5 below. Date: November 13, 2000. The author gratefully acknowledges support by a Forschungsstipendium of the Deutsche Forschungsgemeinschaft. 1

2 STEFAN KEBEKUS The characterization problem arises naturally in the study of projective varieties which are covered by a family of rational curves of minimal degrees: if we are given a projective variety V and a subvariety H Chow(V ) parameterizing a covering family of rational curves of minimal degree, the subfamily H Sing H parameterizing singular rational curves is of greatest interest. It is conjectured that dim H Sing dim V 1. In our previous paper [Keb00] we gave bounds on the dimension of H Sing using the following line of argumentation. First, we chose a general point x V, considered the subfamily Hx Sing H Sing of singular curves which contain x and constructed a partial resolution of singularities as follows: Ũ normalization P 1-bundle U H bundle of singular plane cubics finite morphism finite cover and normalization U π H Sing x Where U Chow(X) X is the universal family and fibers of π are irreducible and generically reduced singular rational curves. Secondly, we noted that the family U comes from the universal family over the Chowvariety and is therefore projective. We were able to show that this is possible only if the parameter space Hx Sing is either finite or if it is 1-dimensional and parameterizes both nodal and cuspidal curves. The argumentation involved an analysis of the intersection numbers of certain divisors in Ũ. A complete description of projective bundles over curves, which is somewhat more delicate and not numerical in nature, has not been given in [Keb00]. In order to complete the picture we discuss it here. It is hoped that these results will be useful in the further study of rational curves on projective varieties. Throughout the present paper we work over the field C of complex numbers and use the standard language of algebraic geometry as introduced in [Har77]. Acknowledgment. The results of this paper were worked out while the author enjoyed the hospitality of RIMS in Kyoto and KIAS in Seoul. The author is grateful to Y. Miyaoka for the invitation to RIMS and to J.-M. Hwang for the invitation to KIAS. He would also like to thank S. Helmke for a number of helpful discussions. 2. RULED SURFACES AND TRANSFORMATIONS OF RULED SURFACES The main results of this paper characterize projective bundles of singular plane cubics by describing their normalizations, which are ruled surfaces. Thus, before stating the results in section 3, it seems advisable to recall some elementary facts about the normalization morphism and about transformations between ruled surfaces. 2.1. Reduction to ruled surfaces. As a first step in the reduction of the characterization problem, we note that to give a bundle of singular plane cubics over a smooth curve C, it is equivalent to give a ruled surface X over C and a double section σ X. Indeed, if a bundle X of singular plane cubics is given, we know from [Kol96, thm. II.2.8] that its normalization will be a P 1 -bundle. The scheme-theoretic preimage of the (reduced) singular locus will be a double section. On the other hand, if π : X C is a P1 -bundle

PROJECTIVE BUNDLES OF SINGULAR PLANE CUBICS 3 containing a double section σ, we can construct a diagram X γ identification X π P 1-bundle C π bundle of singular plane cubics as follows. Find a cover by open subsets U α C so that we can identify π 1 (U α ) = P 1 U α in a way that enables us to write σ π 1 (U α ) = { ([y 0 : y 1 ], x) P 1 U α y0 2 = g(x)y1 2 } where g O(U α ). The identification morphism γ α = γ π 1 (U α) is then locally given as γ α : P 1 U α P 2 U α ([y 0 : y 1 ], x) ( [y 2 0y 1 g(x)y 3 1 : y 3 0 g(x)y 0 y 2 1 : y 3 1)], x ) An elementary calculation shows that the image of γ α has the structure of a bundle of singular plane cubics and that the local morphisms γ α glue together to give a global one. More precisely, for a point µ C with fiber X µ := π 1 (µ), the fiber is nodal if σ intersects X µ := π 1 (µ) in two distinct points and cuspidal if it intersects in a double point. Note that the gluing morphisms do not in general come from automorphisms of P 2. For that reason X can in general not be embedded into a P 2 -bundle over C. 2.2. Elementary transformations. The primary tool in the discussion of ruled surfaces will be the elementary transformation which is a birational map between ruled surfaces. We refer to [Har77, V.5.7.1] for the definition and a brief discussion of these maps and the associated terminology. If π : Y C is a ruled surface and (σ i, D i ) i=1...n is a collection of sections σ i Y and effective divisors D i Div(C) such that the supports D i are mutually disjoint, we can inductively define a birational map between ruled surfaces elt (σi,d i) i=1...n : Y Ỹ as follows. Choose an index j, choose a closed point µ D j and perform an elementary transformation with center π 1 (µ) σ j. Replace the σ i with their strict transforms, replace D i with D i δ ij µ, where δ is the Kronecker symbol, and start anew until all D i are zero. It follows directly from the construction of the elementary transformation that the target variety Ỹ as well as the resulting birational map are independent of the choices made. The inverse of an elementary transformation can again be written as an elementary transformation and the following lemma shows a way to write down the inverse transformation. The proof is very elementary and therefore omitted here. Lemma 2.1. Let π : Y C be a ruled surface and let (σ i ) i=1...n be sections and D i Div(C) be effective divisors with disjoint supports. Assume that for every index i and every point µ D i, there exists a unique index j such that π(d i D j ) µ. Consider the birational map elt (σi,d i) i=1...n : Y Ỹ. If σ i Ỹ are the strict transforms of the σ i, and if we set D i := mult µ (D j ) µ, j i, µ D j \π(σ i σ j) then the inverse map is given as elt 1 (σ i,d i) i=1...n = elt ( σi, D i) i=1...n.

4 STEFAN KEBEKUS 2.3. Projectivity and base change. We will sometimes do a finite base change in order to avoid technical difficulties. It is important to note that this does not make a bundle projective. Lemma 2.2. Let π : X C be a bundle of singular plane cubics and let γ : C C be a finite morphism between smooth curves. Then X := X C C is projective if and only if X is. Proof. It is clear that X is projective if X is, and it remains to consider the other direction. Thus, assume that X is projective. We will show that this implies the projectivity of X. Let f : X X be the natural morphism. Since all fibers of π are generically reduced, it follows from [Har77, Ex. III.10.2] that the singular locus X Sing does not contain π-fibers. Thus, we can always find a reduced, irreducible and very ample Cartier divisor H X such that f(h) intersects the branch locus of f only in smooth points of X note that the branch locus of f is supported on finitely many π-fibers. We are finished if we show that f(h) is Cartier. It follows, however, directly from the construction that f(h) is Cartier in a neighborhood of π-fibers which are in the branch locus of f. On the other hand, if X µ is any other fiber, then we can find a neighborhood U of X µ such that f 1 (U) decomposes into k connected components U (1),..., U (k) such that for all i, the map f U (i) : U (i) U is isomorphic. In this setup it is clear that f(h) U = f(h U (i) ), i=1...k and each summand is Cartier. Hence the claim. 3. CHARACTERIZATION OF PROJECTIVE BUNDLES This section summarizes the main results of this paper. We leave the proofs until after section 4, where the notion of osculating points will be introduced and a number of basic properties proved. 3.1. Bundles of cuspidal curves. For bundles of cuspidal curves, it is particularly simple to characterize those which are projective. Theorem 3.1. Let C be a smooth curve and X C be a bundle of cuspidal plane cubics. If η : X X denotes the normalization, then X is projective if and only if there exists a section σ X over C which is disjoint from the preimage of the singular locus: σ η 1 (X Sing ) =. A proof will be given in section 5. 3.2. Bundles of plane cubics with nodal fibers. Throughout the present section, let C be a smooth curve and π : X C be a bundle of singular plane cubics where not all fibers are cuspidal. Then it follows from deformation theory that all but finitely many fibers are nodal. If η : X X denotes the normalization, then X is a P1 -bundle over C and η 1 (X Sing ) is a reduced double section over C. After performing a finite base change, if necessary, we assume that η 1 (X Sing ) decomposes into two distinct sections σ 0 and σ recall from lemma 2.2 that projectivity of bundles of plane cubics is stable under finite base change. The following is a first, necessary condition for X to be projective. Lemma 3.2. If X is projective, then σ 0 and σ are numerically equivalent as Cartier divisors on X.

PROJECTIVE BUNDLES OF SINGULAR PLANE CUBICS 5 Proof. If X is projective, its Picard-number will be 2, and the quotient Pic( X)/η (Pic(X)) is torsion. It follows that σ 0 and σ are numerically equivalent on X if their intersection with line bundles in η (Pic(X)) agrees. That, however, is clear since η maps both σ 0 and σ birationally onto X Sing. The condition spelled out in lemma 3.2 is, however, not sufficient. For a proper formulation, the following construction is useful. Since X is a P 1 -bundle over C, after performing another base change, we will assume that σ 0 and σ are linearly equivalent and consider the following diagram: (3.1) ˆX elt ( σ0, D 0 ) X η X where D 0 := C C C p σ 0 σ mult p ( σ 0 σ ) π(p ) and mult p ( σ 0 σ ) is the local intersection multiplicity. The map elt ( σ0, D 0) is thus the minimal series of elementary transformations such that the strict transforms σ 0 and σ of σ 0 and σ become disjoint for this, note that it follows immediately from the construction of the elementary transformation that the intersection number between the strict transforms of σ 0 and σ drops exactly by one with each transformation in the sequence elt ( σ0, D 0) until the intersection eventually becomes zero. Observe that σ 0 and σ are still linearly equivalent and that ˆX is therefore isomorphic to the trivial bundle P 1 C. The inverse transformation is described as follows: Theorem 3.3. There exist effective disjoint divisors D i Div(C) and disjoint divisors (σ i ) i=1...n P 1 C which are fibers of the projection P 1 C P 1 such that elt 1 ( σ 0, D 0) = elt (σi,d i) i=1...n. This description finally enables us to formulate necessary and sufficient conditions for X to be projective by characterizing those divisors σ i which come from projective bundles. Theorem 3.4. Let C be a smooth curve, let n be a positive integer, (D i ) i=1...n Div(C) arbitrary disjoint effective divisors and σ 0, (σ i ) i=1...n and σ arbitrary distinct fibers of the projection P 1 C P 1. Construct a bundle X of singular plane cubics as follows. (3.2) P 1 C elt (σ i,d i ) i=1...n projection π 2 elementary transformations π X γ ( σ0, σ ) identification C C C π X bundle of singular plane cubics where γ ( σ0, σ ) is the identification morphism described in section 2.1. The bundle X is projective if and only if there exists a coordinate on P 1 such that σ 0 = {[0 : 1]} C, σ = {[1 : 0]} C and σ i = {[ξ i : 1]} C where the ξ i are roots of unity. The construction (3.2) is of course the inverse of the construction shown in diagram (3.1) above.

6 STEFAN KEBEKUS Remark 3.5. If we fix a base curve C and a number n, then the construction of theorem 3.4 yields a map { } { } n-tuples (Di, σ m : i ) i=1...n isomorphism classes of. of divisors and sections bundles of plane cubics If the curve C has only finitely many automorphisms, the map m will be finite-to-one. In this sense we say that the set P of projective bundles is neither open nor closed in the set M of all bundles of plane cubics. Moreover, every irreducible component P 0 P is a subvariety of a component M 0 M and dim P 0 = 1 2 dim M 0. 4. PROJECTIVE BUNDLES AND OSCULATING POINTS The present section is concerned with an investigation of the special geometry of projective bundles of singular plane cubics. The results will later be used in sections 5 7 to prove the main theorems. Throughout this section we assume that π : X C is a projective bundle of singular plane cubics over a smooth curve C and that L Pic(X) is an ample line bundle. Again let η : X X be the normalization. 4.1. L-osculating points. The restriction of the line bundle L to a fiber defines a number of points which we call L-osculating. More precisely, we use the following definition. Definition 4.1. Let Y be an integral, singular plane cubic, and H Pic k (Y ) be a line bundle of degree k > 0. We call a smooth point σ Y Reg an H-osculating point if O Y (kσ) = H. The following lemma shows how to calculate the H-osculating points on a given curve in a particularly simple situation. Lemma 4.2. Let Y be a nodal plane cubic and p Y Reg be a smooth point. Fix an identification ι : C Y Reg such that ι 1 (p) = 1 and set (σ i ) i=1...k = {ι(ξ) ξ k = 1}. Then σ i are the osculating points for the line bundle O Y (kp). In particular, there exist exactly k osculating points for O Y (kp). Proof. Recall from [Har77, Ex. II.6.7] that the map ι defines a group morphism ι : C Pic 0 (Y ) t O Y (p ι(t)) It follows that O Y (kp) = O Y (kι(t)) if and only if O Y (p ι(t)) k = O Y, i.e. if ι (t) is a kth root of unity. Similarly, we have the following result for cuspidal plane curves. Lemma 4.3. Let Y be a cuspidal plane cubic and H Pic(Y ) be a line bundle of degree k > 0. Then there exists a unique H-osculating point.

PROJECTIVE BUNDLES OF SINGULAR PLANE CUBICS 7 4.2. The variety of L-osculating points. In the setup of this section, the L-osculating points on fibers can be used to define a global (multi-)section σ X. A detailed description of σ will be the key in our argumentation. For this, it is important to note that the relative Picard group is locally divisible. Proposition 4.4. Let k be the relative degree of the line bundle L, i.e. the intersection number of L with a fiber of π. Pick a point µ C, fix a unit disk C centered about µ and set X := π 1 ( ). Then, after shrinking, if necessary, there exists a line bundle L Pic(X ) such that kl = L X. Proof. As a first step, we will prove that H 2 (X, Z) = Z. In order to see this, recall from deformation theory that after shrinking, if necessary X is of the form X = {(x, [y0 : y 1 : y 2 ]) P 2 y 2 y 2 1 y 3 0 f(x)y 2 0y 2 } where f is a function f O( ). In particular, if N X is the non-normal locus and Ñ := η 1 (N) its preimage in the normalization, then N = {[0 : 0 : 1]} is a unit disc and Ñ either a unit disk or a union of irreducible components which are each isomorphic to and meet in a single point. In this setup, we may use the Mayer-Vietoris sequence for reduced cohomology to calculate: 1... H (Ñ, Z) =0 H 2 (X, Z) H 2 ( X, Z) =Z H 2 (N, Z)... =0 See [BK82, prop. 3.A.7, p. 98] for more information about the sequence. Stefan Helmke pointed out that H 2 (X, Z) = Z can also be shown by deforming X into a bundle of cuspidal plane cubics where the claim is obvious. Now choose a section s X which is entirely supported on the smooth locus. After shrinking, this will always be possible. Consider the exponential sequence... H 1 (X, O) α H 1 (X, O ) β H 2 (X, Z) =Z... The element h := (L X O(ks)) satisfies β(h) = 0 and is therefore contained in Pic 0 (X ) = Image(α). Let h α 1 (h) be a preimage and note, that, since H 1 (X, O) is a C-vector space, we can find an element h H 1 (X, O) such that h = k h. We may therefore finish by setting L := α(h ) O(s). The divisibility of L implies that we can locally always find a component of the osculating locus which is contained in the smooth part X Reg X. Corollary 4.5. Fix a point µ C. If C is a sufficiently small unit disk about µ, then there exists an L-osculating section σ 1 X supported in the smooth locus of X. More precisely, there exists a section σ 1 X,Reg such that for all points µ and fibers X µ := π 1 (µ) the intersection σ 1 X µ is an L-osculating point of the fiber X µ. Proof. Let L Pic(X ) be a line bundle such that kl = L; the existence of L is guaranteed by proposition 4.4. Note that R 0 π (L ) is locally free of rank one. Thus, after shrinking if necessary, a section s H 0 (X, L ) exists whose restriction to any fiber of π is not identically zero. But since the relative degree of L is one, the restriction of s to a fiber is a section which vanishes at exactly one smooth point of the fiber. This point must therefore be L-osculating. Thus, the divisor σ 1 L X associated with the section s contains only smooth L-osculating points and maps bijectively onto the base.

8 STEFAN KEBEKUS 4.3. L-osculating points in the presence of a nodal fiber. Throughout this section, let C 0 C be the maximal subset such that all fibers are nodal plane cubics and assume that C 0 is not empty, i.e. assume that nodal fibers exist. Corollary 4.5 already gives a complete description of the L-osculating locus in a neighborhood of a nodal fiber. Lemma 4.6. Let C 0 C be the maximal (open) subset such that all fibers are nodal plane cubics. If k is the degree of the restriction of L to a fiber, then there exists a k-fold unbranched multisection σ XReg 0 X0 := π 1 (C 0 ) such that the restriction to any fiber σ X η is exactly the set of the L-osculating points of that fiber. Proof. Let µ C 0 be any point and C 0 be a small unit disk centered about µ. The preimage X := π 1 ( ) X will then be isomorphic to C N. Let σ 1 X,Reg be the L-osculating section whose existence is guaranteed by corollary 4.5 and find an isomorphism ι : C X,Reg such that σ 1 = ι( {1}). Apply lemma 4.2 to see that σ is then given as Hence the claim. σ = {ι( {ξ i }) ξ k i = 1}. Definition 4.7. In the setup of this section, let σ X be the closure of η 1 (σ ). We call the irreducible components ( σ i ) i=1...n σ the L-osculating (multi-)sections. As a next step we will find coordinates on X := π 1 ( ) where the L-osculating multisection σ can be written explicitly, even if X contains a cuspidal fiber. Proposition 4.8. Assume that the preimage η 1 (X Sing ) consists of two distinct sections σ 0 and σ. If a point p σ 0 σ is given, then there exists a unit disc C about π(p) such that π 1 ( ) σ decomposes into k irreducible components σ i which are sections over. Furthermore, there exists a unique index 1 j k such that p σ j. All other components components σ a, σ b with a, b {0, j, } do contain p. If m = mult p ( σ 0, σ ) is the local intersection multiplicity of σ 0 and σ at p, then mult p ( σ a, σ b ) = mult p ( σ a, σ 0 ) = mult p ( σ a, σ ) = m. Proof. Choose a unit disk C centered about µ := π(p) and equip with a coordinate x. Then X µ := π 1 (µ) is a cuspidal curve. After shrinking we may assume that µ is the only point in whose preimage is cuspidal. By corollary 4.5, we can find an index j such that p σ j. Therefore we can choose a bundle coordinate on X = P1 so that we can write σ 0 = {([y 1 : y 2 ], x) P 1 y 1 = x m y 2 } σ = {([y 1 : y 2 ], x) P 1 y 1 = x m y 2 } σ j = {([y 1 : y 2 ], x) P 1 y 2 = 0} for an integer m > 0. For a point ν, ν µ, the map η ι ν with ι ν : C π 1 (x) t [ν m (t + 1) : (1 t)] parameterizes the smooth part of π 1 (µ). Apply lemma 4.2 with ι := η ι ν and write σ π 1 ( ) = {([y 1 : y 2 ], x) P 1 y 1 (ξ 1) = x m y 2 (ξ + 1), ξ k = 1}. The claim follows.

PROJECTIVE BUNDLES OF SINGULAR PLANE CUBICS 9 5. PROOF OF THEOREM 3.1 Theorem 3.1 is a direct consequence of corollary 4.5. Assume that X is projective. Then, if all fibers of π are cuspidal, the restriction of L to a π-fiber defines a unique osculating point on that fiber, and the local section σ 1 whose existence was shown in corollary 4.5 extends to a global section which is entirely contained in the regular set of X. The existence of σ X follows. On the other hand, if σ is given, then η( σ ) is Cartier and relatively ample. The bundle X will thus be projective. 6. PROOF OF THEOREM 3.3 In the setup of theorem 3.3, it follows from proposition 4.8 that σ. σ 0 = σ. σ = (k 1) σ 0. σ. An elementary calculation shows that the intersection between the strict transforms of σ and σ 0 drops by k 1 with each elementary transformation in the sequence elt ( σ0, D 0). As a direct consequence we obtain that if σ ˆX = P 1 C is the strict transform of σ, then σ, σ 0 and σ are mutually disjoint. It follows that σ decomposes into irreducible components (σ i ) i=1...k which are each fibers of the projection P 1 C P 1. This in turn implies that the L-osculating multisection σ decomposes into k irreducible components which are sections over C. This enables us to apply lemma 2.1 to elt ( σ0, D 0). Actually, the lemma shows that where D i := This ends the proof of theorem 3.3. elt 1 ( σ 0, D 0) = elt (σ i,d i) i=1...n p σ 0 σ, σ i p mult p ( σ 0 σ ) π(p). 7. PROOF OF THEOREM 3.4 7.1. Sufficiency. To begin the proof, we assume that σ i and D i are given as in theorem 3.4 and that there exists a coordinate on P 1 such that the σ i correspond to roots of unity. We will then show that the variety X is projective. More precisely, we fix an index 1 i n and let σ i X be the strict transform of σ i. We will show that the image σ i := γ ( σ0, σ )( σ i ) X is a Q-Cartier divisor. Thus, a suitable multiple of σ i generates a relatively ample line bundle, and we are done. If the construction of theorem 3.4 involves only three sections σ 0, σ 1 and σ, then it is clear that the strict transform σ 1 X of σ 1 is disjoint from the strict transforms σ 0 and σ. Thus, the image σ 1 does not meet the singular locus X Sing of X and is therefore Cartier. If the construction uses more than three sections and σ i is not already Cartier, let µ C be a point such that σ i meets the singular locus X Sing over µ. By construction, there exists a unique index j such that µ D j. It follows that the strict transform σ j of σ j does not intersect σ 0 or σ over µ: π( σ 0 σ j ) µ and π( σ σ j ) µ.

10 STEFAN KEBEKUS We can therefore find a suitable unit disc C centered about p and we can find coordinates x on and a bundle coordinate [y 0 : y 1 ] such that we can write σ 0 = {([y 0 : y 1 ], x) P 1 y 0 = x m y 1 } σ = {([y 0 : y 1 ], x) P 1 y 0 = x m y 1 } σ j = {([y 0 : y 1 ], x) P 1 y 1 = 0} An elementary calculation, using lemma 4.2 and the assumption that there exist coordinates where σ i and σ j are of the form {Root of unity} C shows that σ i = {([y 0 : y 1 ], x) P 1 y 0 = ξ+1 ξ 1 xm y 1 } = {([y 0 : y 1 ], x) P 1 y 0 (ξ 1) + x m y 1 (ξ + 1) = 0} =:f(x,y 0,y 1) where ξ is a root of unity. We fix a number k such that ξ k = 1 and we will show that σ i X is a k-cartier divisor, i.e. k σ i X is Cartier. Recall from section 2.1 that the map γ ( σ0, σ ) is locally given as γ : C C 2 (x, y 0 ) ( x, y 2 0 x 2m, y 0 (y 2 0 x 2m ) ) In particular, the image of γ is isomorphic to Spec R, where γ # (R) k[x, y 0] is the subring generated by the constants C, by x, by y 2 0 and by the ideal (y 2 0 x 2m ); see [Har77, defn. on p. 72] for the notion of γ #. Thus, to show that γ ( σ 0, σ )( σ 1 ) is k-cartier, it suffices to show that f(x, y 0, 1) k γ # (R). We decompose f k as follows. f(x, y 0, 1) k = [(x m y 0 + (x m + y 0 )ξ] k = ( ) k (x m y 0 ) i (x m + y 0 ) k i ξ k i i i=0...k = (x m + y 0 k + (x m y 0 ) k + (x m y 0 )(x m + y 0 )(rest) = ( ) k [x m(k i) y0 i + x m(k i) ( y 0 ) i ] + (x m y 0 )(x m + y 0 )(rest) i i=0...k ( ) k = 2 x m(k i) y0 i (y0 2 x 2m )(rest) i i=0...k, i even =:B =:A It is clear each summand of A is in γ # (R) because it involves only even powers of y 0. Likewise, B γ # (R) as B is contained in the ideal (y2 0 x 2m ). It follows that f k γ # (R), and we are done. 7.2. Necessity. It remains to show that the conditions spelled out in theorem 3.4 are also necessary. For this assume that X is projective. We are finished if we can show that this implies the existence of a coordinate on P 1 such that σ 0, σ and σ i correspond to [0, 1], [1, 0] and [ξ i, 1] for certain roots of unity ξ i. To accomplish this, choose a general point µ C. The fiber X µ := π 1 (C) will then be a nodal curve. By Riemann-Roch, there exists a point p X µ such that O Xµ (kp) = L Xµ (kp). It follows directly from lemma 3.1 that we find coordinates on X µ = η 1 (X µ ) such that σ 0 X µ corresponds to [0 : 1], σ X µ corresponds to [1 : 0], and the L- osculating points correspond to [ξ i : 1], where ξ i are roots of unity. Note that the rational map elt ( σ0, D 0) is an isomorphism in a neighborhood of X µ and use the coordinates on X µ

PROJECTIVE BUNDLES OF SINGULAR PLANE CUBICS 11 to obtain a global bundle coordinate on ˆX = P 1 C = X µ C. This coordinate will have the desired properties, and the proof is finished. REFERENCES [BK82] G. Barthel and L. Kaup. Sur la Topologie des Surfaces complexes compactes, chapter in Topologie des surfaces complexes compactes singulières, pages 61 297. Number 80 in Semin. Math. Super. Les Presses de l université de Montréal, 1982. [Har77] R. Hartshorne. Algebraic Geometry, volume 52 of Graduate Texts in Mathematics. Springer, 1977. [Keb00] S. Kebekus. Families of singular rational curves. LANL-Preprint math.ag/0004023, 2000. [Kol96] J. Kollár. Rational Curves on Algebraic Varieties, volume 32 of Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge. Springer, 1996. INSTITUT FÜR MATHEMATIK, UNIVERSITÄT BAYREUTH, 95440 BAYREUTH, GERMANY E-mail address: stefan.kebekus@uni-bayreuth.de URL: http://btm8x5.mat.uni-bayreuth.de/ kebekus