On symplectic fillings of small Seifert $3$-manifolds

In this paper, we investigate the minimal symplectic fillings of small Seifert 3-manifolds with a canonical contact structure. As a result, we classify all minimal symplectic fillings of small Seifert 3-manifolds satisfying certain conditions. Furthermore, we also demonstrate that every such a minimal symplectic filling is obtained by a sequence of rational blowdowns from the minimal resolution of the corresponding weighted homogeneous complex surface singularity.


Introduction
One of the fundamental problems in symplectic 4-manifold topology is to classify symplectic fillings of certain 3-manifolds equipped with a natural contact structure.Among them, researchers have long studied symplectic fillings of the link of a normal complex surface singularity.Note that the link of a normal surface singularity carries a canonical contact structure also known as the Milnor fillable contact structure.For example, P Lisca [8], M Bhupal and K Ono [1], and H Park, J Park, D Shin and G Urzúa [12] completely classified all minimal symplectic fillings of lens spaces and certain small Seifert 3-manifolds coming from the link of quotient surface singularities.L Starkston [16] also investigated minimal symplectic fillings of the link of some weighted homogeneous surface singularities.
On the one hand, topologists working on 4-manifold topology are also interested in finding a surgical interpretation for symplectic fillings of a given 3-manifold.More specifically, one may ask whether there is any surgical description of those fillings.In fact, it has been known that rational blowdown surgery, introduced by R Fintushel and R Stern [5] and generalized by the second author [14] and A Stipsicz, Z Szabó and J Wahl [18], is a powerful tool to answer this question.For example, for the link of quotient surface singularities equipped with a canonical contact structure, it was proven that every minimal symplectic filling is obtained by a sequence of rational blowdowns from the minimal resolution of the singularity; see Bhupal and Ozbagci [2], Choi and Park [4].On the other hand, L Starkston [17] showed that there are symplectic fillings of some Seifert 3-manifolds that cannot be obtained by a sequence of rational blowdowns from the minimal resolution of the singularity.Note that Seifert 3-manifolds can be viewed as the link of weighted homogeneous surface singularities.Hence, it is an intriguing question as to which Seifert 3-manifolds have a rational blowdown interpretation for their minimal symplectic fillings.
In this paper, we investigate the minimal symplectic fillings of small Seifert 3-manifolds over the 2-sphere satisfying certain conditions.By a small Seifert (fibered) 3-manifold, we assume that it admits at most 3 singular fibers when it is considered as an S 1fibration over the 2-sphere.In general, a Seifert 3-manifold as an S 1 -fibration over a Riemann surface can have any number of singular fibers.We denote a small Seifert 3-manifold Y by Y .bI .˛1;ˇ1/; .˛2;ˇ2/; .˛3;ˇ3// whose surgery diagram is given in Figure 1 and which is also given as a boundary of a plumbing 4-manifold of disk bundles of 2-spheres according to the graph in Figure 1.The integers b ij 2 in Figure 1 are uniquely determined by the continued fraction ˛i ˇi D OEb i1 ; b i2 ; : : : If the intersection matrix of a plumbing graph is negative definite, which is always true for b 3, then there is a canonical contact structure on Y induced from a symplectic structure of the plumbing 4-manifold, where each vertex corresponds to a symplectic 2-sphere and each edge represents an orthogonal intersection between the symplectic 2-spheres; see Gay and Stipsicz [7].Furthermore, the canonical contact structure on Y is contactomorphic to the contact structure defined by the complex tangency of a complex structure on the link of the corresponding singularity, which is called the Milnor fillable contact structure; see Park and Stipsicz [13].
This paper aims to classify all possible list of minimal symplectic fillings of small Seifert 3-manifolds satisfying certain conditions, and to prove that every such a minimal symplectic filling is obtained by a sequence of rational blowdowns from the minimal resolution of the corresponding weighted homogeneous surface singularity, as it is true for a quotient surface singularity.Our strategy is as follows.For a given minimal symplectic filling W of Y .bI .˛1;ˇ1/; .˛2;ˇ2/; .˛3;ˇ3// with b 4, we glue W with a concave cap K to get a closed symplectic 4-manifold X .Then, since the concave cap K always contains an embedded .C1/ 2-sphere corresponding the central vertex, X is a rational symplectic 4-manifold by McDuff [9].Furthermore, the adjunction formula and intersection data impose a constraint on the homological data of K in X Š CP 2 ]N CP 2 .Under blowdowns along all exceptional 2-spheres away from the .C1/ 2-sphere in X Š CP 2 ]N CP 2 , the concave cap K becomes a neighborhood of symplectic 2-spheres which are isotopic to b number of complex lines through symplectic 2-spheres in CP 2 ; see Starkston [16; 17] for details.Since the symplectic deformation type of W Š X nK is determined by the isotopy class of a symplectic embedding of K within a fixed homological embedding, we investigate a symplectic embedding of K using a curve configuration corresponding to W , which consists of strands representing irreducible components of K and exceptional 2-spheres intersecting them (Definition 3.1 and Figure 5).Since the curve configuration corresponding to W determines a symplectic embedding of K, we can recover all minimal symplectic fillings by investigating all possible curve configurations of Y .Sometimes, we find a certain chain of symplectic 2-spheres lying in W , which can be rationally blowing down, from the homological data of K.Note that by rationally blowing down the chain of symplectic 2-spheres lying in W , we obtain another minimal symplectic W 0 from W .In this case, we keep track of changes in the homological data of K so that we get a curve configuration of W 0 from that of W . Finally, by analyzing the effect of rational blowdown surgery on the curve configuration of minimal symplectic fillings, we obtain our main result.
Theorem 1.1 For a small Seifert 3-manifold Y .bI .˛1;ˇ1/; .˛2;ˇ2/; .˛3;ˇ3// with its canonical contact structure and b 4, all minimal symplectic fillings of Y are listed explicitly by curve configurations.Furthermore, they are also obtained by a sequence of rational blowdowns from the minimal resolution of the corresponding weighted homogeneous surface singularity.
Remark L Starkston [16] originally described a general scheme for how to obtain minimal symplectic fillings from given homological data of an embedding of the cap and got some results in special cases.L Starkston [17] also showed that the isotopy type of a symplectic line arrangement is uniquely (up to deformation equivalence) determined by its intersection data in the cases that multi-intersection points of a symplectic line arrangement satisfy some mild conditions, which contain the cases appearing in Proposition 3.4.Thus, by combining Propositions 3.3 and 3.4 with Starkston's result [17,Proposition 4.2], we conclude that there exists at most one minimal symplectic filling for each possible curve configuration.Then we prove in Section 4 that every such a curve configuration gives the corresponding minimal symplectic filling, which implies the first statement in Theorem 1.1 above.
Acknowledgements The authors would like to thank the referees for their valuable comments.Jongil Park was supported by Samsung Science and Technology Foundation under Project Number SSTF-BA1602-02 and by the National Research Foundation of Korea (NRF) Grant 2020R1A5A1016126 funded by the Korea government.He also holds a joint appointment in the Research Institute of Mathematics, SNU.

Weighted homogeneous surface singularities and Seifert 3-manifolds
We briefly recall some basics of weighted homogeneous surface singularities and Seifert 3-manifolds; see [10] for details.Suppose that .w0 ; : : : ; w n / are nonzero rational numbers.A polynomial f .z0 ; : : : ; z n / is called weighted homogeneous of type .w0 ; : : : ; w n / if it can be expressed as a linear combination of monomials Equivalently, there exist nonzero integers .q0 ; : : : ; q n / and a positive integer d satisfying f .tq 0 z 0 ; : : : t q n z n / D t d f .z0 ; : : : ; z n /.Then, a weighted homogeneous surface singularity .X; 0/ is a normal surface singularity that is defined as the zero loci of weighted homogeneous polynomials of the same type.Note that there is a natural C -action given by t .z0 ; : : : ; z n / D .tq 0 z 0 ; : : : ; t q n z n /; with a single fixed point 0 2 X .This C -action induces a fixed-point-free S 1 C action on the link L WD X \ @B of the singularity, where B is a small ball centered at the origin.Hence, the link L is a Seifert fibered 3-manifold over a genus g Riemann surface, denoted by Y .bI gI .˛1;ˇ1/; .˛2;ˇ2/; : : : ; .˛k;ˇk// for some integers b, ˛i and ˇi with 0 < ˇi < ˛i and .˛i;ˇi/ D 1.Note that k is the number of singular fibers, and there is an associated star-shaped plumbing graph : the central vertex has genus g and weight b, and each vertex in k arms has genus 0 and weight b ij uniquely determined by the continued fraction ˛i ˇi D OEb i1 ; b i2 ; : : : For example, Figure 1 shows the case of g D 0 and k D 3, which is called a small Seifert (fibered) 3-manifold.By P Orlik and P Wagreich [11], it is well known that the plumbing graph is a dual graph of the minimal resolution of .X; 0/.Conversely, if the intersection matrix of is negative definite, there is a weighted homogeneous surface singularity whose dual graph of the minimal resolution is ; see [15].Note that a Seifert 3-manifold Y , as a boundary of a plumbed 4-manifold according to , inherits a canonical contact structure providing that each vertex represents a symplectic 2-sphere, all intersections between them are orthogonal, and the intersection matrix of is negative definite; see [7].Furthermore, if the Seifert 3-manifold Y can be viewed as the link L of a weighted homogeneous surface singularity, then the canonical contact structure above is contactomorphic to the Milnor fillable contact structure, which is given by T L \ J T L; see [13].

Rational blowdowns and symplectic fillings
Rational blowdown surgery, first introduced by R Fintushel and R Stern [5], is one of the most powerful cut-and-paste techniques.It replaces a certain linear plumbing C p of disk bundles over a 2-sphere whose boundary is a lens space L.p 2 ; p 1/ with a rational homology 4-ball B p which has the same boundary.Later, Fintushel and Stern's rational blowdown surgery was generalized by J Park [14] using a configuration C p;q obtained by linear plumbing disk bundles over a 2-sphere according to the dual resolution graph of L.p 2 ; pq 1/, which also bounds a rational homology 4-ball B p;q .
In the case of a symplectic 4-manifold .X; !/, rational blowdown surgery can be performed in the symplectic category: if all 2-spheres in the plumbing graph are symplectically embedded and their intersections are !-orthogonal,then the surgered 4-manifold X p;q D .X C p;q / [ B p;q also admits a symplectic structure induced from the symplectic structure of X ; see [19; 20].In fact, the rational homology 4-ball B p;q admits a symplectic structure compatible with the canonical contact structure on the boundary L.p 2 ; pq 1/.More generally, in addition to the linear plumbing of 2-spheres, there is a plumbing of 2-spheres according to star-shaped plumbing graphs with 3 or 4 legs admitting a symplectic rational homology 4-ball; see [18; 3].That is, the corresponding Seifert 3-manifold Y .b; .˛1;ˇ1/; .˛2;ˇ2/; .˛3;ˇ3//, or Y .b; .˛1;ˇ1/; .˛2;ˇ2/; .˛3;ˇ3/; .˛4;ˇ4// with a canonical contact structure, has a minimal symplectic filling whose rational homology is isomorphic to that of the 4-ball [6].For example, a plumbing graph p;q;r in Figure 3 can be rationally blown down.We will use this later in the proof of the main theorem.
As rational blowdown surgery does not affect the symplectic structure near the boundary, if there is a plumbing of disk bundles over symplectically embedded 2-spheres that can be rationally blown down, then one can obtain another symplectic filling by replacing the plumbing with a rational homology 4-ball.In the case of the link of quotient surface singularities, it was proven [2; 4] that every minimal symplectic filling is obtained by a sequence of rational blowdowns from the minimal resolution of the singularity, which is diffeomorphic to a plumbing of disk bundles over symplectically embedded 2-spheres.First, they constructed a genus-0 or genus-1 Lefschetz fibration X on each minimal symplectic filling of the link of a quotient surface singularity.Suppose that w 1 and w 2 are two words consisting of right-handed Dehn twists along curves in a generic fiber, which represent the same element in the mapping class group of the generic fiber.If the monodromy factorization of X is given by w 1 w 0 , one can construct another Lefschetz fibration X 0 whose monodromy factorization is given by w 2 w 0 .The operation of replacing w 1 with w 2 is called a monodromy substitution.Next, they showed that the monodromy factorization of each minimal symplectic filling of the link of a quotient surface singularity is obtained by a sequence of monodromy substitutions from that of the minimal resolution.Furthermore, these monodromy substitutions can be interpreted as rational blowdown surgeries topologically.Note that all rational blowdown surgeries mentioned here are linear: a certain linear chain C p;q of 2-spheres is replaced with a rational homology 4-ball.

Minimal symplectic fillings of a small Seifert 3-manifold
In this subsection, we briefly review Starkston's results [16; 17]  For a given minimal symplectic filling W of Y , we glue W and K along Y to get a closed symplectic 4-manifold X .Then, the existence of a .C1/ 2-sphere implies that X is a rational symplectic 4-manifold and, after a finite number of blowdowns, X becomes CP 2 and the .C1/ 2-sphere in K becomes a complex line CP 1 CP 2 ; see McDuff [9] for details.Under these circumstances, it is natural to ask the following question: What is the image of K in CP 2 under blowdowns?In the case that K is linear, which means that the corresponding Y is a lens space, Lisca showed that the image of K is two symplectic 2-spheres in CP 2 , each of which is homologous to CP  By analyzing the proof of Lisca's result [8, Theorem 4.2], Starkston [16] showed that the image of K is b symplectic 2-spheres in CP 2 , each of which is homologous to CP 1 CP 2 .For the complete classification of minimal symplectic fillings of Y , one needs to classify the isotopy classes of these b symplectic 2-spheres, which are called symplectic line arrangements.Since all these spheres are J -holomorphic for some J tamed by the standard Kähler form of CP 2 and are homologous to CP 1 CP 2 , they intersect each other at a single point for each pair of 2-spheres.Note that these intersection points need not be all distinct.These intersection data of a symplectic line arrangement are determined by the homological data of K, which also have constraints from the adjunction formula.In [17], Starkston showed that symplectic line arrangements with certain types of intersections are isotopic to complex line arrangements, that is, the corresponding b symplectic 2-spheres are isotopic (through symplectic spheres) to b complex lines in CP 2 .For example, Starkston classified minimal symplectic fillings by an explicit computation of all possible homological embeddings of K for some families of Seifert fibered spaces; see [16, Sections 3 and 4.4; 17, Section 5].

Strategy for the main theorem
As we saw in the previous section, for each minimal symplectic filling W of Y , we obtain a rational symplectic 4-manifold X which is symplectomorphic to CP 2 ]N CP 2 for some integer N by gluing K to W along Y .Conversely, given an embedding of a concave cap K into CP 2 ] N CP 2 , we obtain a symplectic filling W of Y by taking a complement of K in CP 2 ] N CP 2 .So the classification of minimal symplectic fillings of Y is equivalent to the classification of the embeddings of K into CP 2 ] N CP 2 for some N .Hence, in order to investigate minimal symplectic fillings W of Y , we first introduce two notions, homological data and curve configuration of the corresponding embedding of K, which are defined as follows.
Definition 3.1 Suppose that W is a minimal symplectic filling of a small Seifert 3-manifold Y equipped with a concave cap K. Then we have an embedding of K into a rational symplectic 4-manifold X Š CP 2 ] N CP 2 such that the .C1/ 2-sphere in K is identified with CP 1 CP 2 .Let l be a homology class represented by a complex line CP 1 in CP 2 and e i be homology classes of exceptional spheres coming from blowups.Then fl; e 1 ; : : : ; e N g becomes a basis for H 2 .X I Z/, so that the homology class of each irreducible component of K can be expressed in terms of this basis, which we call the homological data of K for W .
Note that K is symplectically embedded in X Š CP 2 ]N CP 2 and each irreducible component of K can be assumed to be J -holomorphic for some J tamed by standard Kähler form on X .Then there is a sequence of blowdowns from X to CP 2 and we can find a Jholomorphic exceptional sphere † i whose homology class is e i disjoint from the central .C1/ 2-sphere of K at each stage of blowing down.Because of the J -holomorphic condition and homological restrictions from the adjunction formula together with intersection data of K, the exceptional sphere † i intersects positively at most once with the image of an irreducible component of K or is one of the image of irreducible components of K; see [8,Proposition 4.4].In particular, for each image C j of irreducible components of K, the intersection number between e i and OEC j lies in f 1; 0; 1g.
As mentioned in the previous section, we finally get a symplectic line arrangement in CP 2 which consists of J -holomorphic 2-spheres, each of which is the image of the first component of each arm under blowdowns.The intersection data of the symplectic line arrangement are determined by the homological data of K, so that it can be represented as a configuration of strands: each strand represents a J -holomorphic 2-sphere of a symplectic line arrangement in CP 2 , while the intersection of two strands represents a geometric intersection of two 2-spheres.Then, starting from the configuration of the symplectic line arrangement, we can draw a configuration C of strands with degrees by blowups according to the homological data of K until we get K in the configuration.Here the degree of each strand in C means a self-intersection number of the strand.To be more precise, when we blow up a point p on a strand in a configuration, we introduce a new strand with degree 1 to the point p so that we resolve intersection of strands at p and we decrease the degree of the strands containing p by one.Hence the configuration C , which represents the total transform of a symplectic line arrangement, contains strands representing irreducible components of K and exceptional .1/ 2-spheres intersecting with the irreducible components.We say that two configurations C 1 and C 2 for W are equivalent if there is a bijective map between .1/ strands preserving intersections with the irreducible components of K. of K. We denote the exceptional 2-spheres by dash-dotted strands.See Figure 5 for example.
Remark We often use the terminology configuration of strands when we deal with an intermediate configuration between a symplectic line arrangement and a curve configuration, or a configuration containing K but with strands with degree less than or equal to 2 other than irreducible components of K.
Proposition 3.3 For given homological data of K for W , there is a unique curve configuration C up to equivalence.
Proof Since each strand in a curve configuration C represents a J -holomorphic 2-sphere for some J tamed by standard Kähler form on X Š CP 2 ] N CP 2 , all intersections between the strands represent positive geometric intersections between the corresponding J -holomorphic 2-spheres.Note that there is at most one intersection point between any two strands due to homological restrictions.Furthermore, if e i is a homology class of an exceptional 2-sphere satisfying e i OEC j 2 f0; 1g for any irreducible component C j of K, then there is a .1/ strand L i in C whose homology class is e i ; otherwise, there is a blowup on the strand L i so that the proper transform of L i becomes an irreducible component C j of K whose intersection with e i is 1, contradicting the assumption.Hence there is a .1/ strand L i representing a J -holomorphic exceptional sphere † i whose homology class is e i in C if and only if e i OEC j 2 f0; 1g for any irreducible component C j of K.
Let C and C 0 be two curve configurations for a fixed homological data of K for W .Then, the numbers of .1/ strands in C and C 0 are equal to the number of e i satisfying the condition e i OEC j 2 f0; 1g for any irreducible component C j of K. Hence we can construct a desired bijection between the .1/ strands by finding correspondence between such e i and .1/ strands in two curve configurations.Now, we investigate minimal symplectic fillings of a given small Seifert 3-manifold Y by analyzing all the possible curve configurations.For this, we first determine all possible symplectic line arrangements.
Proof Since Y is a small Seifert 3-manifolds with b 4, we can always choose a concave cap K with a .C1/ central 2-sphere and b 4 arms, each of which consists of a single .1/ 2-sphere as in Figure 4. Furthermore, since the blowdowns are disjoint from the central 2-sphere in K, each of b 1 arms in K descends to a single .C1/ J -holomorphic 2-sphere intersecting at a distinct point with an image of the central 2-sphere of K under the blowdowns.Let C 1 ; C 2 ; : : : ; C b 4 be the images of b 4 many .1/ 2-spheres in K under the blowdowns.Then they should have a common intersection point in CP 2 : otherwise, we have distinct two points p and q on some C i such that C i intersects C j and C k at p and q, respectively.Let r be an intersection point of C j and C k .Then any J -holomorphic 2-sphere coming from an arm of K other than C 1 ; : : : C b 4 must pass two of p; q and r , which is a contradiction.
If b 6, a similar argument shows that there is at most one J -holomorphic 2-sphere coming from an arm of K intersecting at a different point from the common intersection point p with C i , which proves the proposition.In the case of b Ä 5, we can easily check that Figure 6 gives all possible symplectic line arrangements: if b D 5, then there is only one C 1 coming from .1/ 2-sphere from K. Recall that there are at most two intersection points on C 1 .If there is only one intersection point on C 1 , then we get the left-hand figure in Figure 6.If there are two intersection points p and q on C 1 , then two of three J -holomorphic 2-spheres coming from the arms of K other than C 1 pass p, and the other passes q (or vice versa), so that we get the right-hand figure in Figure 6.For b D 4 case, we have only three strands in a figure for a symplectic line arrangement except the strand from .C1/ 2-sphere so that we have only two possibilities.
Next, for the complete classification of minimal symplectic fillings of Y , we need to consider the isotopy classes of embeddings of K with a fixed homological data in  6 are actually isotopic to complex line arrangements through symplectic configurations, which means that there is a unique minimal symplectic filling up to symplectic deformation equivalence for each possible choice of homological data of K. Since a choice of homological data of K gives a unique curve configuration C up to equivalence by Proposition 3.3, we analyze minimal symplectic fillings of a small Seifert 3-manifold Y by considering all possible curve configurations obtained from the complex line arrangements in Figure 6.
As previously mentioned, in the case of quotient surface singularities that include all lens spaces and some small Seifert 3-manifolds, every minimal symplectic filling is obtained by linear rational blowdown surgeries from the minimal resolution of the corresponding singularity.However, this is not true anymore for small Seifert 3-manifolds in general.For example, a rational homology 4-ball of p;q;r in Figure 3 might not be obtained by linear rational blowdown surgeries.Nevertheless, many cases such as b 5 are in fact obtained by linear rational blowdowns from their minimal resolutions.For the case of b D 4, one might need 3-legged rational blowdown surgeries to get a minimal symplectic filling.Hence, it is natural to prove the two cases b 5 and b D 4 separately.

The case b 5
We consider all possible curve configurations coming from two complex line arrangements in Figure 6 which can be divided into three types.First, we need to blow up all intersection points in the line arrangements so that we get two configurations as in Figure 8.There are two possibilities for a strand representing exceptional sphere in intermediate configurations coming from blowups: to blow up some intersection points, or not.Once we blow up an intersection point on a strand representing an exceptional sphere †, which means the proper transform of † becomes an irreducible component of K, we should blow up all the intersection points except one intersection point because each strand intersecting the strand for † become irreducible components of distinct arms in K.We can also blow up the last intersection point we did not blow up to get another curve configuration, but it is not necessary in general.
If we do blow up an intersection point on the dash-dotted strand of the left-hand side of Figure 8, we get the configuration on the left-hand side of Figure 9.When we start with two configurations in Figure 9, we can assume without loss of generality that the first three arms become essential arms in K, which consist of strands with degree less than or equal to 2. Since the degree of the other arms is already 1, we can only blow up e 1 and e 2 among dotted exceptional strands.In conclusion, we can divide all the possible curve configurations into following three types: Type A Curve configurations obtained from Figure 8, left, without blowing up the exceptional strand.Type B Curve configurations obtained from Figure 9, left or right, by blowing up at most one e i , with 1 Ä i Ä 2.
Type C Curve configurations obtained from Figure 9, left or right, by blowing up both e 1 and e 2 .

The case b D 4
We divide all curve configurations for b D 4 into the following two cases: Curve configurations of types A, B or, C as in the b 5 case.
Type D Curve configurations obtained from Figure 9, right, by blowing up all exceptional .1/ strands.
Then, since we can deal with the first case using the same argument in the b 5 case, it suffices to prove the type D case whose corresponding curve configurations come from some configurations C p;q;r in Figure 22, which are obtained from the right-hand figure in

Proof of main theorem
In this section, for a given possible curve configuration C , we show that there is a sequence of rational blowdowns from the minimal resolution z M to the minimal symplectic filling W of Y corresponding to C .Since any minimal symplectic filling of a lens space is obtained by a sequence of rational blowdowns from a linear plumbing which is the minimal resolution corresponding to the lens space [2], it suffices to construct a sequence of curve configurations C D C 0 ; C 1 ; : : : ; C n such that each minimal symplectic filling W i corresponding to C i is obtained from W iC1 by replacing a certain linear plumbing L i with its minimal symplectic filling.Here C n denotes a curve configuration for the minimal resolution z M .As previously mentioned, since our possible symplectic line arrangements are isotopic to complex line arrangements, it suffices to work in complex category with a symplectic form ! coming from the standard Kähler form on CP 2 .In order to show that there is a symplectic embedding of L i in W iC1 , we construct a configuration C 0 iC1 of strands, which is not a curve configuration for W iC1 , from a complex line arrangement by blowups with the same homological data of K for W iC1 so that we have L i disjoint from K in C 0 iC1 .Since we work in complex category, each strand in C 0 iC1 can be considered as a complex rational curve in a rational surface X while the intersections between strands represent positive geometric intersections between the corresponding rational curves.This observation implies that L i is symplectically embedded in W iC1 .Now we introduce the notion of standard blowups, which frequently appears in the construction of W i from W iC1 .Let K and K 0 be two star-shaped plumbing graphs having the same number of arms together with a .C1/ central vertex, and let a ij for 1 Ä j Ä n i and a 0 ij for 1 Ä j Ä n 0 i be the weights (equivalently, degrees) of the j th vertex in the i th arm of K and K 0 , respectively.We say K 0 Ä K if n 0 i Ä n i and a 0 ij Ä a ij for any i and j except for a 0 i n 0 i < a i n 0 i in the case of n 0 i < n i .The condition K 0 Ä K guarantees that we can obtain a configuration of strands representing K by blowups from a configuration representing K 0 in the following way: we blow up nonintersection points of the last component of each i th arm in K 0 consecutively until we get n i components, and then we blow up each component at nonintersection points to get the right weights.Definition 4.1 Let C 0 be a configuration of strands obtained from a complex line arrangement by blowups containing a star-shaped plumbing graph K 0 with a homological data.If K 0 Ä K and the degree of all strands in C 0 n K 0 is 1, then we can obtain a curve configuration z C 0 from C 0 by blowing up at nonintersection points only.In this case, we say that the curve configuration z C 0 is obtained by standard blowups from C 0 .
Remark With given homological data of K 0 in C 0 , the standard blowups induce a unique homological data of K for z C 0 : Let e be a homology class of an exceptional sphere coming from blowing-ups from C 0 to z C 0 .Since we blow up nonintersection points, e appears in at most two OEC i 1 j 1 and OEC i 2 j 2 , where C i j denotes the j th component in the i th arm of K.Moreover, if e appears in two OEC i 1 j 1 and OEC i 2 j 2 , then i 1 D i 2 D i and j 2 D j 1 C 1 with e OEC i j 1 D 1 and e OEC i j 1 C1 D 1.
For a given star-shaped plumbing graph K 0 Ä K, in general if n 0 i < n i for some i , where n 0 i and n i are the number of components in i th arm of K 0 and K respectively, there are possibly other ways of blowing up to get the i th arm of K from that of K 0 .Let C 0 be a configuration of strands containing K 0 Ä K as in Definition 4.1.Assume furthermore that n 0 i < n i for some i .Let z C 0 be a curve configuration obtained from C 0 by standard blowups.Then we get the following three fundamental lemmas.Lemma 4.2 Let C be a curve configuration for K, and let W be the minimal symplectic filling of Y corresponding to C .Suppose C 0 is a configuration for K 0 Ä K such that the standard blowups z C 0 of C 0 differs from C only in the components C i j for n 0 i Ä j Ä n i .Let z W denote the minimal symplectic filling of Y corresponding to z C 0 .Then there is a symplectically embedded linear plumbing L of 2-spheres determined by OEb 1 ; b 2 ; : : : ; b r in z W such that W is obtained by z W by replacing the plumbing L with some minimal filling W L of the lens space boundary of the linear plumbing L. Further, OEb 1 ; b 2 ; : : : ; b r is the dual of OE.a i n 0 i a 0 i n 0 i /; a i n 0 i C1 ; a i n 0 i C2 ; : : : ; a i n i , where a ij and a 0 ij are the weights of the j th component in the i th arm of K and K 0 , respectively.

Hakho Choi and Jongil Park
Proof We can assume that a i n 0 i a 0 i n 0 i 2 because the way of blowing up from the i th arm of K 0 to that of K remains the same when we replace K 0 with K 00 , where K 00 is obtained from K 0 by blowing up the last component of the i th arm.
First we show that there is a symplectic linear embedding L in z W .Let S be a configuration of strands containing K obtained as follows.We blow up the last component in the i th arm of K 0 in C 0 at a nonintersection point so that we have two consecutive strands of degree a 0 i n 0 i 1 and 1.Since the continued fraction OEb 1 ; b 2 ; : : : ; b r is dual to OE.a i n 0 i a 0 i n 0 i /; a i n 0 i C1 ; a i n 0 i C2 ; : : : a i n i by the definition of L, we obtain a linear chain of strands containing the rest of the i th arm in K and L from the two strands by blowing up consecutively at intersection points as in Figure 10, so that there is an embedding L in the complement of K in a rational surface X .Furthermore, since we started from the same homological data of K 0 in C 0 and since a blowup for C 0 to S either increases the number of components or decreases the degree of an irreducible component of K, the homological data of K for both z C 0 and S are the same, so that there is a symplectic embedding L in z W .
Before we examine the effect of replacing L with its minimal symplectic filling W L , we briefly review the classification of minimal symplectic fillings of lens space, which can be found in [1] and [8].For notational convenience, we denote a linear plumbing graph and a lens space determined the plumbing graph by the same L.For a lens space L given by OEb 1 ; b 2 ; : : : ; b r , we can choose a concave cap K L of the form where OEa 1 ; a 2 ; : : : ; a n is a dual continued fraction of OEb 1 ; b 2 ; : : : Let OEC i j C and OEC i j C 0 be homology classes of C i j in C and C 0 respectively.Note that C is a curve configuration completed from the last .a 0 i n 0 i / strand in the i th arm of K 0 by blowups without using any other strand in C 0 .If we blow up in the same ways starting with a single .C1/ strand instead of .a 0 i n 0 i / strand, we get a curve configuration C W L containing K L .Hence there is a minimal symplectic filling , where e D fe 1 ; : : : ; e N g is homology classes of exceptional spheres coming from the blowups from C 0 to C .Finally, we show that the minimal symplectic filling W corresponding to C is given by .z W n L/ [ W L .Suppose X 0 is a rational symplectic 4-manifold obtained by blowups from a complex line arrangement so that it contains C 0 .We take a small Darboux neighborhood B 0 of a disk D in C i n 0 i of K 0 so that B 0 is disjoint from any other irreducible components of K 0 .Now we arrange all the blowups from C 0 to C inside B 0 and let B be blowups of B 0 .Then we have a symplectic embedding of K in X D .X 0 n B 0 / [ B and homological data of K that agrees with C .Furthermore, B n K is symplectic deformation equivalent to W L : consider two complex lines in CP 2 and a symplectic embedding of B 0 such that the image of D in C i n 0 i is a disk in one complex line and B 0 is disjoint from the other complex line.By the construction of B, there is a symplectic embedding of where the first component of K L is the complex line in .CP 2  n B 0 / and the complement of Assume furthermore that there is a .1/ curve intersecting both C i n 0 i and another irreducible component C k l of K 0 in C 0 .Then there is a slight modification of the Lemma 4.2, involving two arms of K.
and components C i j for n 0 i Ä j Ä n i , then there is a symplectically embedded linear plumbing L W , described in Figure 11, such that W is obtained by z W by replacing the plumbing L with some minimal filling W L .Furthermore, OEb 1 ; b 2 ; : : : i C1 ; a i n 0 i C2 ; : : : ; a i n i , where a ij and a 0 ij are the weights of the j th component in the i th arm of K and K 0 , respectively.
Proof The proof is similar to that of Lemma 4.2 except for blowups at two intersection points of E in C 0 to find an embedding L. That is, we construct a configuration S of strands containing K, as in Figure 12, whose homological data is equal to that of z C 0 , so that there is a symplectic embedding of L in z W .
Next, by viewing L as a two-legged plumbing graph with a degree .b 1 1/ of a central vertex, we get a concave cap K L as in Figure 11: starting from the zero section and infinity section together with two generic fibers of F b 1 1 , we construct arms corresponding to OE 2; : : : ; 2 and OE b 2 ; : : : ; b r .Then, by consecutive blowups at intersection points of the proper transform of zero section and the arm corresponding to OE b 2 ; : : : ; b r , we get a concave cap K L for L. As before, for a given minimal symplectic filling of L, we get a rational symplectic 4-manifold by gluing K L along L and the image of K L in CP 2 under blowing down is three complex lines in CP 2 intersecting generically, implying that any curve configuration for K L is obtained from blowing up at an intersection point between two complex lines in CP 2 .Therefore, using blowup data from C 0 to C (Figure 13), we get a minimal symplectic filling W L of L.
Suppose that X 0 is a rational symplectic 4-manifold containing C 00 , obtained from C 0 by blowing down E, and let B 0 be a small Darboux neighborhood of the intersection point coming from the blowing down.Then a similar argument as in Lemma 4.2 above shows that the minimal symplectic filling W corresponding to C is obtained from z W by replacing L with W L .
: : : Assume that C 0 is a curve configuration containing K 0 Ä K corresponding to a minimal symplectic filling W 0 of another small Seifert 3-manifold Y 0 and z C 0 is a curve configuration obtained from C 0 by standard blowups.Then we can describe a minimal symplectic filling z W of Y corresponding to z C 0 explicitly.
Lemma 4.4 Under the assumption above, there is a symplectically embedded plumbing of 2-spheres 0 in the minimal resolution z M such that a minimal symplectic filling z W of Y corresponding to z C 0 is obtained from z M by replacing 0 with W 0 .
Proof Let K 0 be a plumbing graph determined by black strands in the left-hand side of Figure 8.Clearly, K 0 Ä K so that there is a curve configuration C z M obtained by standard blowups from Figure 8, left.We first show that the curve configuration C z M corresponds to the minimal resolution z M .Recall that a concave cap K in Figure 4 can be found in [18; 16]: Starting from the zero and infinity sections with b 1 generic fibers of a Hirzebruch surface F 1 which can be drawn as the left-hand side in Figure 8, we blow up intersection points of generic fibers and the infinity section so that we have a .b/ rational curve which corresponds to the central vertex of the minimal resolution graph .Then, we obtain a linear chain of strands containing both i th arm of K and from two .1/ strands by blowups, as in Figure 14, bottom.As a result, we have a configuration S z M containing both and K disjointly, so that the complement of K in a rational surface X Y is the minimal resolution z M and K is a concave cap for Y .By : : : : : : using the same argument as in the proof of Lemma 4.2 above, we conclude that C z M is a curve configuration for z M .
In the same way, we could get a configuration S 0 of strands containing both K 0 and a plumbing graph 0 so that the complement of K 0 in the resulting rational symplectic 4-manifold X Y 0 Š CP 2 ] M CP 2 is a plumbing of 2-spheres according to 0 .Note that M 1 is the number of blowups in the standard blowups from the left-hand side of Figure 8 to K 0 .Since K 0 Ä K, we obtain a configuration S 0 z M of strands containing 0 and K disjointly from S 0 by standard blowups at nonintersection points in the last component of each i th arm of K 0 .Let X D X Y 0 ] N CP 2 be a resulting rational symplectic 4-manifold.Then X Š X Y D z M [ K, since the number of blowups in the standard blowups from the left-hand side of Figure 8 to K is equal to the sum of numbers of blowups for the left-hand side of Figure 8 to K 0 , and K 0 to K. Furthermore, the homological data of K in S 0 z M is also equal to that of C z M .Hence a plumbing graph 0 is symplectically embedded in z M .
If there is a sequence of blowups from a configuration of strands representing K 0 to K, then we have a corresponding symplectic cobordism Z from Y 0 to Y because the total transform of K 0 is still a concave cap for Y 0 while K is a concave cap for Y .
In particular, if C 0 is a curve configuration for a minimal symplectic filling W 0 , then we get a curve configuration C for a minimal filling W by the sequence of blowups from K 0 to K and W D W 0 [ Z.In the case of standard blowups from K 0 to K, we can deduce from the construction of S 0 z M that the corresponding cobordism is equal to z M n 0 .Hence we have z W D W 0 [ .z M n 0 /, so that z W is obtained from z M by replacing 0 with W 0 .

Proof for type A
For a curve configuration C of type A, we want to show that the corresponding minimal symplectic filling W is obtained from the minimal resolution z M by replacing each arm in the resolution graph with its minimal symplectic filling.Since we already know in the proof of Lemma 4.4 above that a curve configuration C z M , which is obtained from the left-hand side of Figure 8 by standard blowups, corresponds to z M by repeatedly applying Lemma 4.2 with K 0 as in the left-hand side of Figure 8 so that the corresponding L is one of three arms in , we conclude that all minimal symplectic fillings corresponding to a curve configuration C of type A are obtained by a sequence of rational blowdowns from the minimal resolution z M .
The following example illustrates this case.
Example 4.1 Let Y be a small Seifert 3-manifold whose associated plumbing graph and concave cap are shown in Figure 15, top.Then, there are two curve configurations of type A as in Figure 15, bottom.Of course, there exist other curve configurations of type B and C for minimal symplectic fillings of Y , which will be treated in Example 4.2 and Example 4.3 later.Note that each dash-dotted strand represents an exceptional 2sphere, that is, a 2-sphere with self-intersection 1.We omit the degree of irreducible components of the concave cap for the sake of convenience in the figure.The bottom left curve configuration in Figure 15 is obtained by standard blowups from that of Figure 8, which means that the corresponding minimal filling is the minimal resolution.Note that only the third arm in the plumbing graph has a nontrivial minimal symplectic filling that is obtained by rationally blowing down the .4/ 2-sphere.Using Lisca's description of the minimal symplectic fillings of lens spaces, we obtain the bottom right curve configuration in Figure 15, which represents a minimal symplectic filling obtained from the minimal resolution by rationally blowing down the .4/ 2-sphere in the third arm.

Proof for type B
For a curve configuration C of type B, we want to show that the corresponding minimal symplectic filling W is obtained from the minimal resolution z M by replacing disjoint subgraphs in the resolution graph with their minimal symplectic filling.By reindexing if needed, we assume that the first and the second arms of the configurations in Figure 9 become the first and the second arm of K in C , respectively, and the proper transform of e 2 is not an irreducible component of K. Since we do not use e 2 during the blowups, we can get the first and the second arm of K, so that the homological data for the irreducible components in these arms agrees with that of C , from the configurations in Figure 9 leaving the third single .1/ arm unchanged.Hence we arrange the order of blowups from a configuration in Figure 9 to C so that we have an intermediate configuration C 0 of strands containing K 0 Ä K as in Figure 16.Note that the degree of strands in C 0 n K 0 is all 1.If we choose a linear plumbing graph to be a subgraph of as a two-legged plumbing graph with the .b/ central vertex, then K 0 gives a concave cap of L 0 and C 0 is a curve configuration for a minimal symplectic filling W L 0 of L 0 .
Let C 1 be a curve configuration obtained by standard blowups from C 0 .Then, by Lemma 4.4, the curve configuration C 1 corresponds to a minimal symplectic filling W 1 , which is obtained from the minimal resolution z M by replacing L 0 with W L 0 .Furthermore, since OEa 31 ; a 32 ; : : : ; a 3n 3 D OE2; : : : ; 2; c 1 C 1; c 2 ; : : : ; c k , where OEc 1 ; c 2 ; : : : ; c k  of with their minimal symplectic fillings, so that there is a sequence of rational blowdowns from z M to W .
The following example illustrates the curve configurations of type B. Since the left-hand configuration without exceptional 2-spheres in Figure 17 gives a concave cap of a lens space determined by a subgraph of , it gives a minimal symplectic filling W L of the lens space L.39; 16/.Then, by blowups at points lying on the third arm different from the intersection point with the exceptional curve e, we get an embedding of a concave cap K of Y as in the right-hand curve configuration C 1 of Figure 17, which gives a minimal symplectic filling W 1 of Y .Furthermore, since there is a unique minimal symplectic filling of lens space L.2; 1/ corresponding to the .2/ 2-sphere in the third arm of , W 1 is obtained from the minimal symplectic filling W L .In fact, there are three more minimal symplectic fillings of Y which are of type B -see Figure 17 for the corresponding curve configurations.Note that the curve configuration C 1 for W 1 in Figure 17 comes from the right-hand configuration in Figure 9 and the curve c becomes a component of the first arm of K in the top right of Figure 15.Similarly, the curve configuration C i for W i .2Ä i Ä 4/ is also obtained from the right-hand configuration in Figure 9.One can easily check that each W i is obtained from the minimal resolution of Y by a linear rational blowdown surgery: explicitly, W 2 , W 3 and W 4 are obtained by rationally blowing down along subgraphs in , respectively.And W 1 is also obtained by rationally blowing down along which is embedded in another plumbing

Proof for type C
For a minimal symplectic filling W corresponding to a curve configuration C of type C, we want to find a curve configuration C 1 of type B such that there is a symplectically embedded linear chain L of 2-spheres (that is not visible in ) in W 1 corresponding to C 1 so that W is obtained from W 1 by replacing L with its minimal symplectic filling W L .By reindexing if needed, we may assume that the first and the second arm of configurations in Figure 9 become that of K respectively, and the proper transform of e 2 becomes an irreducible component in the third arm of K after blowups.For convenience, we omit in figures all exceptional .1/ strands that intersect only one irreducible component of the corresponding concave cap K. Now, by blowups at intersection points of e 1 consecutively, we get the first and the second arm so that the homological data for the irreducible components in these arms agrees with that of C except for one irreducible component, say C 1 n , of the first arm of K leaving the third single .1/ arm unchanged.Note that there is only one exceptional strand e connecting the first and the second arm as in Figure 18 because we blow up at intersection points of e 1 to get the first and the second arm of K. Hence we can arrange a sequence of blowups from a configuration in Figure 9 to a curve configuration C of type C so that we have an intermediate configuration C 0 of strands as in Figure 18: The left-hand/right-hand figures comes from the left-hand/right-hand figures in Figure 9, respectively.For simplicity, we only explain a curve configuration coming from the left-hand side in Figure 9.In contrast to the type B case, we have a .a 0 1n / strand with a 1n > a 0 1n in C 0 because we need to blow up at the intersection point of e 2 and c in Figure 9, which becomes the .a 1n / strand in the curve configuration C at the top of Figure 19.Let C 1 be a curve configuration obtained from C 0 by standard blowups and W 1 be a minimal symplectic filling of Y corresponding to C 1 .Then, by Lemma 4.3, there is a symplectic embedding L in W 1 so that W is obtained from W 1 by replacing L with its minimal symplectic filling W L , where L is a plumbing graph at the bottom of Figure 19.Since a curve configuration C 1 for W 1 is of type B, there is a sequence of rational blowdowns from z M to W as desired.20.The curve configuration C 5 is obtained from the right-hand configuration in Figure 9, and the proper transforms of e 1 and e 2 are irreducible components of the concave cap K. Thus, as in the proof, we can find an intermediate configuration C 0 between the right-hand configuration in Figure 9 and C 5 .Then it is easy to check that the homological data for standard blowups z C 0 of C 0 and that of C 1 is equal; see Figures 17 and 20.From the proof of Lemma 4.3, we  Here e i and E i denote the homology classes of exceptional spheres in X L and X 0 L .Note that the homological data of L in X L is given by e e 5 e 6 .Therefore, if we see X as X L ] 4CP

Proof for type D
We start to prove this case for a curve configuration coming from C 0;0;0 .Note that C 0;0;0 itself is a curve configuration containing K 0;0;0 corresponding to rational homology ball filling of 0;0;0 in Figure 3.By repeatedly blowing up at intersection points between exceptional strands and the first component of each arm, we can get a curve configuration C p;q;r containing K p;q;r corresponding to a rational homology ball filling of p;q;r as in Figure 22.For notational convenience, we denote three exceptional strands in each C p;q;r by the same e i with i 2 Z 3 so that e i intersects the last component of i th arm and the first component of .iC1/ st arm of K p;q;r .Let C 1; 1; 1 be the right-hand figure of Figure 9 and C p;q; 1 be a configuration of strands obtained from C p;q;0 by blowing down e 2 in Figure 22.Then C p;q; 1 contains K p;q; 1 , which is the proper transform of K p;q;0 under blowing down.Proof Since there are no strands with degree Ä 2 in C except for irreducible components of K, each irreducible component of K 0;0;0 in C 0;0;0 should become an irreducible component of K under blowups from C 0;0;0 to C .Hence, in order to get C from C 0;0;0 by blowups e i , we should blow up at either two intersection points of e i with arms or an intersection point of e i with the .iC1/ st arm only.Note that we get C p;q;r containing K p;q;r by blowups the latter case repeatedly.Hence, by rearranging the order of blowups from C 0;0;0 to a curve configuration C , we may assume that C is obtained from C p;q;r with p; q; r 0 and there are no more blowups at an intersection point of e i with the .iC1/ st arm only.Since the configuration C p;q;r clearly satisfies conditions (i) and (ii), we are done if there is no blowup at intersection points of K p;q;r in C p;q;r .
If there are blowups at intersection points of K p;q;r in C p;q;r to C , then we will find another C a;b;c with a p, b q and c r satisfying conditions (i)-(iii) as follows.Let x i be the first intersection point in the i th arm of K p;q;r among the intersection points to be blown up, and C 0 be a configuration of strands obtained by blowing up at x i for 1 Ä i Ä 3.For notational convenience, we denote exceptional strands in C p;q;r and the proper transform of e i in C 0 by the same e i .There is a unique .1/ exceptional strand in each i th arm of K 0 in C 0 , which is the n i th component of the i th arm with n i 2, where K 0 is the total transform of K p;q;r .Then we claim that there is a sequence of blowups from C n 1 3;n 2 3;n 3 3 to C 0 : we blow up two intersection points of e i simultaneously, and then we blow up at the intersection point between the exceptional .1/ strand and the first component of the .iC1/ st arm consecutively to get C 0 ; for example, see Figure 23 for the first arm.We see from the construction that a configuration C n 1 3;n 2 3;n 3 3 satisfies conditions (i) and (ii).Moreover, since x i is We end this section by giving an example of minimal symplectic fillings involving 3-legged rational blowdown surgery.
Example 4.4 Let Y be a small Seifert 3-manifold whose minimal resolution graph and concave cap K are given by Figure 24.We consider two minimal symplectic fillings W 1 and W 2 of Y whose respective curve configurations are given by Figure 25, top and bottom.Note that the curve configuration in Figure 25, top, is obtained from C 0;0;0 by standard blowups.Thus, as in the proof, W 1 is obtained from the minimal resolution by rationally blowing down 0;0;0 .Let us denote v 0 by a central vertex and v j i by i th vertex of the j th arm in .Then, v 0 ; v 1 1 ; v 2 1 and v 3 1 C v 3 2 give a symplectic embedding of 0;0;0 to the minimal resolution.A computation similar to that of Example 4.3 shows that there is a symplectic embedding L of

3r 3 Figure 1 :
Figure 1: Surgery diagram of Y and its associated plumbing graph .

Figure 7 :
Figure 7: Configuration for C i , C j and C k .

Figure 9 ;
see Section 4.4 for details.The main difference between the b D 4 case and the b 5 case is that one can use all three exceptional 2-spheres to get a concave cap K for b D 4, while one can use only e 1 and e 2 for b 5 from the right-hand figure in Figure 9.

Figure 10 :
Figure 10: Finding an embedding of L.

Figure 11 :
Figure 11: A plumbing graph of L and its concave cap K L .

kl 1 Figure 12 :
Figure 12: Embedding of L to z W .

Figure 13 :
Figure 13: Top: blowups from C 0 to C .Bottom: curve configuration for W L .

Figure 14 :
Figure 14: Top: blowing up Hirzebruch surface F 1 .Bottom: construction of each arm in K and .

Figure 15 :
Figure 15: Top: plumbing graph , and its concave cap K. Bottom: two curve configurations in Example 4.1.

Example 4 . 2
We again consider a small Seifert 3-manifold Y used in Example 4.1.

Figure 17 :
Figure 17: Top: curve configuration C 1 for W 1 .Bottom: curve configurations for other symplectic fillings of Y .

Figure 20 :L 1 D 5 2 to W 1
Figure 20: Curve configuration C 5 for symplectic filling W 5 of Y .

Figure 21 :
Figure 21: Top: embedding of L 1 in W 1 .Bottom: two curve configurations for Y L .

Figure 24 :
Figure 24: Plumbing graph and its concave cap K.
The condition b 4 on the weight (equivalently, degree) of a central vertex of the plumbing graph ensures that one can always choose a concave cap K, which is also star-shaped, with a .C1/ central 2-sphere and b 4 arms, each of which consists of a single .1/2-sphereas in Figure4.Here OEa i1 ; a i2 ; : : : ; a i n i denotes a dual continued fraction of OEb i1 ; b i2 ; : : : ; b i r i ; that is, ˛i=.˛i ˇi/ D OEa i1 ; a i2 ; : : : ; a i n i while ˛i=ˇi D OEb i1 ; b i2 ; : : : ; b i r i .
a rational symplectic 4-manifold obtained by gluing two plumbings according to L and K L whose second homology class is generated by flg [ E D fE 1 ; : : : ; E N 0 g.Then, for a given minimal symplectic filling W L of L, we get a rational symplectic 4-manifold X W L Š CP 2 ] N CP 2 by gluing W L and K L and the image of K L under blowing down is isotopic to two complex lines in CP 2 , which means that a minimal symplectic filling of L is determined by a choice of homological data of K L in CP 2 ] N CP 2 for some N .Hence, we draw a curve configuration C W L for W L starting from a configuration of two .C1/ strands in CP 2 by blowing-ups with only one .C1/ strand.This observation shows that the effect of replacing L in X L with W L is the following: We have another rational symplectic 4-manifold X W L Š CP 2 ] N CP 2 and the second homology classes in the complement of L are changed so that l !l andOEL i E !OEL i e for 1 Ä i Ä n;where OEL i E and OEL i e are homology classes of irreducible components of K L in terms of flg [ E D fE 1 ; : : : ; E N 0 g and flg [ e D fe 1 ; : : : ; e N g respectively.
where C 0 is the central .C1/ 2-sphere of K, l is the homology class representing the complex line in CP2, and e i is the homology class of each exceptional 2-sphere.As in the proof of Lemma 4.3, we can find a symplectic embedding of X whose homological data is given by e 3 e 5 e 7 e 8 e 9 and e 9 e 10 ; refer to Figure21, top.There are two minimal symplectic fillings of L whose corresponding curve configurations are as in Figure21, bottom.Note that the first figure represents a linear plumbing while the second figure represents a rational homology 4-ball.Hence, if we rationally blow downL from X L D L [ K L Š CP 2 ] 6CP 2 , then we get a new rational symplectic 4-manifold X 0 L Š CP 2 ] 4CP 2 ,and the homological data of K L changes as follows: l !l; l e 1 e 2 !l E 1 E 2 ; e 2 e 3 !E 2 E 3 ; e 3 e 4 !E 3 E 4 ; e 4 e 5 e 6 !E 1 E 2 E 3 : 8 e 9 e 10 !E 1 E 2 E 3 ;where e 1 ; e 2 ; e 4 ; e 6 and E 1 ; E 2 ; E 3 ; E 4 represent the standard exceptional 2-spheres in X 0 Š CP 2 ] 8CP 2 .Therefore, the new homological data for the concave cap K, which give the right-hand curve configuration in Figure20, are as follows:Remark We investigated all possible curve configurations for a small Seifert 3manifold Y with b 5 in the proof of the main theorem.As a result, we can find all minimal symplectic fillings of Y via corresponding curve configurations.For example, a complete list of minimal symplectic fillings of Y in Example 4.1 are given by Examples 4.1-4.3.
2, we get X 0 Š CP 2 ] 8CP 2 by rationally blowing down L from X , and the homological data of K L is changed by l !l; l e 3 e 5 !lE 1 E 2 ;e 5 e 7 !E 2 E 3 ;e 7 e 8 !E 3 E 4 ; e