FREE AND NON-FREE MULTIPLICITIES ON THE A 3 ARRANGEMENT

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1 FREE AND NON-FREE MULTIPLICITIES ON THE A 3 ARRANGEMENT MICHAEL DIPASQUALE, CHRISTOPHER A. FRANCISCO, JEFFREY MERMIN, AND JAY SCHWEIG Abstact. We give a complete classification of fee and non-fee multiplicities on the A 3 baid aangement. Namely, we show that all fee multiplicities on A 3 fall into two families that have been identified by Abe-Teao-Wakefield (007) and Abe-Nuida-Numata (009). The main tool is a new homological obstuction to feeness deived via a connection to multivaiate spline theoy. 1. Intoduction Let V = K l be a vecto space ove a field K of chaacteistic zeo. A cental hypeplane aangement A = {H 1,..., H n } is a set of hypeplanes H i V passing though the oigin in V. In othe wods, if we let {x 1,..., x l } be a basis fo the dual space V and S = Sym(V ) = K[x 1,..., x l ], then H i = V (α Hi ) fo some choice of linea fom α Hi V, unique up to scaling. A multi-aangement is a pai (A, m) of a cental aangement A and a map m : A Z 0, called a multiplicity. If m 1, then (A, m) is denoted A and is called a simple aangement. The module of deivations on S is defined by De K (S) = l i=1 S x i, the fee S-module with basis xi = / x i fo i = 1,..., l. The module De K (S) acts on S by patial diffeentiation. Ou main object of study is the module D(A, m) of logaithmic deivations of (A, m): D(A, m) := {θ De K (S) : θ(α H ) α m(h) H fo all H A}, whee α m(h) H S is the ideal geneated by α m(h) H. If D(A, m) is a fee S-module, then we say (A, m) is fee o m is a fee multiplicity of the simple aangement A. Fo a simple aangement, D(A, m) is denoted D(A); if D(A) is fee we say A is fee. The module of logaithmic deivations is cental to the theoy of hypeplane aangements, initiated and studied by Saito in [Sai75, Sai80]. In paticula, it is impotant to know when A is a fee aangement. Indeed, possibly the most impotant open question in hypeplane aangements is whethe feeness is a combinatoial popety; see, fo instance, [OT9]. Yoshinaga [Yos04] has shown that feeness of an aangement is closely elated to feeness of the canonical esticted multi-aangement defined by Ziegle [Zie89]. Hence the feeness of multiaangements is impotant to the theoy of hypeplane aangements as well. The baid aangement of type A l is defined as {H ij = V (x i x j ) : 0 i < j l} in V = K l+1. Fee multiplicities on baid aangements have been studied in [Te0, ST98, AY09, Yos0, ANN09]. Until ecently thee have been vey few tools to study multi-aangements. In two papes [ATW07, ATW08], Abe-Teao-Wakefield extend the theoy of the chaacteistic polynomial and deletion-estiction aguments to 1

2 M. DIPASQUALE, C.A. FRANCISCO, J. MERMIN, AND J. SCHWEIG multi-aangements. These allow new methods fo detemining the feeness (and non-feeness) of multiaangements. In paticula, the tool of local and global mixed poducts is intoduced fo chaacteizing non-feeness of multi-aangements in some instances. Abe [Abe07] uses these tools to give the fist non-tivial complete classification of fee and non-fee multiplicities on a hypeplane aangement, the so-called deleted A 3 aangement. The main esult of this pape is the next natual step; namely a complete chaacteization of fee and non-fee multiplicities on the A 3 baid aangement. Thee ae two main classes of multiplicities that have been chaacteized as fee on the A 3 baid aangement. The fist class may be descibed as follows. Suppose that, fo some index i, the inequalities m(h jk ) m(h ij )+m(h ik ) 1 ae satisfied fo evey pai of distinct indices j i, k i (geometically, thee hypeplanes which intesect in codimension two have elatively high multiplicity compaed to the othe thee hypeplanes). If these inequalities ae satisfied, we say that the index i is a fee vetex fo m. If m has a fee vetex, then it is known that m is a fee multiplicity [ATW08, Coollay 5.1] (see also Coollay 3.17). To descibe the second (much moe complex) class of fee multiplicities, take fou non-negative integes n 0, n 1, n, and n 3 and conside the multiplicity m(h ij ) = n i + n j + ɛ ij, whee ɛ ij { 1, 0, 1}. We call these ANN multiplicities, due to a classification of all such multiplicities as fee o non-fee by Abe, Nuida, and Numata in [ANN09]. It tuns out the multiplicity m(h ij ) = n i + n j is always fee, and the classification of all ANN multiplicities depends on measuing the deviation fom these using signed-eliminable gaphs. We descibe this classification in moe detail in Section 6. Ou main esult is that all fee multiplicities on A 3 fall into these two classes. Theoem 1.1. The multi-baid aangement (A 3, m) is fee if and only if m has a fee vetex o m is a fee ANN multiplicity. We pove Theoem 1.1 via a connection to multivaiate splines fist noted by Schenck in [Sch14] and futhe developed by the fist autho in [DiP16]. Ou main tool, Theoem 3.16, is a new citeion fo feeness of a multi-baid aangement (A 3, m) in tems of syzygies of ideals geneated by powes of the linea foms defining the hypeplanes of A 3. This condition gives a obust obstuction to feeness which we use to establish Theoem 1.1. Ou pape is aanged as follows. In Section, we intoduce the notation and backgound we will use thoughout the pape. Section 3 uses homological techniques to pove Theoem 3.16, which says that the multi-aangement (A 3, m) is fee pecisely when a cetain syzygy module is locally geneated. Reades may safely skip the est of that section and simply ead the theoem statement if they desie. In Sections 4 and 5, we pove Theoem 1.1 using Theoem 3.16 along with combinatoial aguments using syzygies and Hilbet functions. In Section 6, we ecove the non-fee multiplicities in the classification of Abe-Nuida-Numata [ANN09]. We conclude with emaks on using the fee ANN multiplicities of [ANN09] to constuct minimal fee esolutions fo cetain ideals geneated by powes of linea foms. In Appendix A, we illustate the classification of Theoem 1.1 in the case of two-valued multiplicities.. Notation and peliminaies In this section we set up the main notation to be used thoughout the pape. The data of the A 3 aangement is captued in a labeling of the vetices of K 4,

3 FREE AND NON-FREE MULTIPLICITIES ON THE A 3 ARRANGEMENT 3 the complete gaph on fou vetices; namely the edge between v i and v j in K 4 coesponds to the hypeplane H ij = V (x i x j ). As such we will also denote A 3 by A K4. Put m ij = m(h ij ). We will ecod the multiplicities of the hypeplanes as a lexicogaphically odeed list m = (m 01, m 0, m 03, m 1, m 13, m 3 ) which we can also associate to the obvious labelling of the edges of K 4. We will often efe to the multiplicities as a, b, c, d, e, f accoding to the edge-labeling in Figue 1. v 1 e a d v 3 c v 0 f b v Figue 1. Labelling Convention Fo simplicity, we set S = K[x 0, x 1, x, x 3 ] and α ij = x i x j fo all i > j. Ou goal is to study when the module of multi-deivations D(A 3, m) = {θ de K (S) : θ(α ij ) α mij ij fo all 0 i < j 3} is fee as an S-module. Remak.1. Note that thee is a line contained in evey hypeplane of A 3, namely the line descibed paametically as {(t, t, t, t) : t K}. Thus A 3 is not essential; an essential aangement is one in which all hypeplanes intesect in only the oigin. Pojecting along this line we obtain an aangement in K 3 whose hypeplanes may be descibed as follows. Set x = x 1 x 0, y = x x 0, z = x 3 x 0. Then the essential A 3 aangement in K 3 is A e 3 = {V (x), V (y), V (z), V (y x), V (z x), V (z y)}. See Figue fo a pictue of this aangement in R 3. Set R = K[x, y, z]. It is not difficult to see that D(A 3, m) = D(A e 3, m) R S. Hence feeness of (A 3, m) and (A e 3, m) ae equivalent. We will suppess the distinction between A e 3 and A 3, calling both the A 3 aangement. We will also suppess the distinction between the polynomial ings S = K[x 0, x 1, x, x 3 ] and R = K[x, y, z], simply letting S efe to the ambient polynomial ing in both situations. It will be obvious fom context (but not impotant) which polynomial ing is meant. In the next section, which is the technical heat of the pape, we will show that feeness of D(A 3, m) is detemined by syzygies of cetain ideals which we now define. Fo any edge e = {i, j}, we set the ideal J(ij) = α mij ij. Moe geneally, fo any subset σ {0, 1,, 3}, we set J(σ) = J(ij). {i,j} σ Fo instance, J(01) = J(01) + J(0) + J(1). Using x, y, z in place of x 1 x 0, x x 0, x 3 x 0 as in Remak.1 and the multiplicity labels (a, b, c, d, e, f) as in Figue 1,

4 4 M. DIPASQUALE, C.A. FRANCISCO, J. MERMIN, AND J. SCHWEIG Figue. Essential A 3 aangement the following is a list of all ideals J(σ) fo σ {0, 1,, 3}. J(01) = x a J(01) = x a, y b, (x y) d J(0) = y b J(013) = x a, z c, (x z) e J(03) = z c J(03) = y b, z c, (y z) f J(1) = (x y) d J(13) = (x y) d, (x z) e, (y z) f J(13) = (x z) e J(013) = x a, y b, z c, (x y) d, (x z) e, (y z) f J(3) = (y z) f Theoem 3.16 will show that the feeness of the multi-aangement (A 3, m) depends on the elationship between the global fist syzygy module syz(j(013)) and its local fist syzygies syz(j(ijk)), fo 0 i < j < k Technical machiney The bulk of this section is technical, and the goal is simply to pove Theoem Late sections equie only the statement of this theoem, so eades wishing to avoid the technical details can safely skip to Section 4.1. In paticula, additional notation intoduced in this section is not used elsewhee in the pape. A gaphic aangement is a subaangement of a baid aangement. Moe pecisely, let G be a vetex-labeled gaph on l+1 vetices {v 0,..., v l } with no loops o multiple edges. Denote by E(G) the set of edges of G. We denote the edge between vetices v i, v j by {i, j}. The gaphic aangement coesponding to G is A G = V (x j x i ) K l+1. {i,j} E(G) A gaphic multi-aangement (A G, m) is a gaphic aangement A G with an assignment m : E(G) N of a positive intege m(e) to evey edge e E(G) Homological necessities. Ou main tool to study feeness of D(A G, m) is a chain complex R/J [G] whose top homology is the module D(A G, m), intoduced in [DiP16]. We now define this complex. Denote by (G) the clique complex of G. This is the simplicial complex on the vetex set of G whose simplices ae given by sets of vetices that induce a complete subgaph (clique) of G. Denote by (G) i the set of cliques of G with (i+1) vetices, i.e., the simplices of (G) of dimension i.

5 FREE AND NON-FREE MULTIPLICITIES ON THE A 3 ARRANGEMENT 5 Definition 3.1. Let G be a gaph with l + 1 vetices and set S = K[x 0,..., x l ]. Define the complex R[G] to be the simplicial co-chain complex of (G) with coefficients in S; that is, R[G] i = S[e γ ], whee [e γ ] is a fomal symbol coesponding to the i-dimensional clique γ. The diffeential δ i : R[G] i R[G] i+1 is γ (G) i the simplicial diffeential of the co-chain complex of (G) with coefficients in S. Remak 3.. By definition, H (R[G]) is isomophic to the cohomology of (G) with coefficients in S. In the following definition, if e = {i, j} E(G), we will denote α ij = x i x j by α e. Definition 3.3. Let (A G, m) be a gaphic multi-aangement. Let σ = {j 0, j 1,..., j i } be a clique of G. Then If σ is a vetex of G, then J(σ) = 0. J(σ) := αe me e E(σ). Definition 3.4. Given a gaphic multi-aangement (A G, m), J [G] is the subchain complex of R[G] with J [G] i = J(γ)[e γ ]. R/J [G] denotes the quotient γ (G) i complex R[G]/J [G] with R/J [G] i = (S/J(γ))[e γ ]. γ (G) i Lemma 3.5. The module of multi-deivations D(A G, m) of the gaphic multiaangement (A G, m) is H 0 (R/J [G]). Poof. Let F R[G] 0. Wite F = (..., F v,...) v V (G). Then F ke( δ 0 ) if and only if, fo all e = {i, j} E(G), we have (δ(f )) e = F i F j J(e) = (x i x j ) m(e). This last statement is the definition of D(A G, m). With Lemma 3.5 as ou justification, we will call R/J [G] the deivation complex of G. Example 3.6. Take G to be the thee-cycle with labeling as in Figue 3. v σ 1 v 0 v 1 Figue 3. Thee-cycle fo Example 3.6 The shot exact sequence of complexes 0 J [G] R[G] R/J [G] 0 is shown below.

6 6 M. DIPASQUALE, C.A. FRANCISCO, J. MERMIN, AND J. SCHWEIG δ 0 δ 1 J [G] 0 J(01) J(0) J(1) J(01) 0 δ 0 δ 1 R[G] 0 S 3 S S S S 0 R/J [G] 0 S 3 S J(01) S J(0) S J(1) δ 0 δ 1 The diffeentials ae δ 0 = δ 1 ( = ) S J(01) The homologies H i (R[G]) vanish fo i = 1,, and H 0 (R[G]) = S. The coesponding long exact sequence in (co)homology splits up to yield the shot exact sequence 0 S H 0 (R/J [G]) H 1 (J [G]) 0, and an isomophism H 1 (R/J [G]) = H (J [G]) = 0. The shot exact sequence actually splits, so H 0 (R/J [G]) = S H 1 (J [G]). The map δ 1 : J(01) J(0) J(1) J(01) is sujective by definition, hence H (R/J [G]) = 0. Also, H 1 (J [G]) = ke(δ 1 ) = syz(j(01)), the module of syzygies on J(01). Hence H 0 (R/J [G]) = D(A G, m) = S syz(j(01)). Remak 3.7. The ideal J(01) in Example 3.6 is codimension two and Cohen- Macaulay. Hence D(A G, m) = S syz(j(01)) is a fee module egadless of the choice of m 01, m 0, m 1. It is well-known that ank two aangements ae totally fee fo the same eason; they ae second syzygy (o eflexive) modules of ank two. Remak 3.8. In Example 3.6, we undestand syz(j(01)) to epesent syzygies among the geneatos (x 0 x 1 ) m01, (x 1 x ) m1, (x 0 x ) m0, even if this is not a minimal geneating set. Fo instance, if m 01 + m 1 m 0 + 1, then J(01) is geneated by (x 0 x 1 ) m01 and (x 1 x ) m1. In this case, syz(j(01)) is geneated by the Koszul syzygy on (x 0 x 1 ) m01, (x 1 x ) m1 and the elation of degee m 0 expessing (x 0 x ) m0 as a polynomial combination of (x 0 x 1 ) m01, (x 1 x ) m1. In Example 3.6, H i (R/J [G]) = 0 fo i = 1,, and D(A G, m) was fee. This is no coincidence. Theoem 3.9. [DiP16, Theoem 3.] The gaphic multi-aangement (A G, m) is fee if and only if H i (R/J [G]) = 0 fo all i > 0. Theoem 3.9 follows fom a esult of Schenck using a Catan-Eilenbeg spectal sequence [Sch97]. Although we use Theoem 3.9 in this pape pimaily to study the multi-baid aangements (A 3, m), we show in the following example how it may be used to classify fee multiplicities on othe gaphic aangements. Example 3.10 (Deleted A 3 aangement). Conside the gaph G in Figue 4. This is the simplest example of a gaph whee feeness of (A G, m) depends on the multiplicities m. 0

7 FREE AND NON-FREE MULTIPLICITIES ON THE A 3 ARRANGEMENT 7 v 1 (z x) e x a (y x) d v 0 v 3 z c y b v Figue 4. Gaph fo the deleted A 3 aangement The maps in cohomology (the diffeentials of R[G]) ae given by δ 0 = δ 1 = ( ) , whee the ows and columns ae labeled by faces (see Figue 4). Let us set x = x 1 x 0 = α 01, y = x x 0 = α 0, z = x 3 x 0 = α 03. Then α 3 = x 3 x = z y and α 13 = x 3 x 1 = z x. Suppose the edge {i, j} is assigned multiplicity m ij. Set m 01 = a, m 0 = b, m 03 = c, m 1 = d, m 13 = e. We have J(01) = x a, y b, (y x) d J(013) = x a, z c, (z x) e. Remak The following chaacteization of fee multiplicities on the deleted A 3 aangement in Example 3.10 is deived in [Abe07] using techniques fo multiaangements developed in [ATW07, ATW08]. We show how this chaacteization may be obtained homologically fom Theoem 3.9. Poposition 3.1. Let A G be the deleted A 3 aangement fom Example With notation as in Example 3.10, (A G, m) is fee if and only if eithe c+e a+1 o b + d a + 1. Poof. We have H i (R[G]) = 0 fo i > 0 since (G) is contactible, and H 1 (R/J [G]) = H (J [G]) via the long exact sequence coesponding to 0 J [G] R[G] R/J [G] 0. The complex J [G] has the fom J(ij) δ1 J(013) J(03). {i,j} G The map δ 1 is given by the matix δ 1 = ( ) Let us detemine when δ 1 is sujective, hence when H (J [G]) = H 1 (R/J [G]) = 0. We see that, given (f 1, f, f 3, f 4, f 5 ) J [G] 1, δ 1 (f 1, f, f 3, f 4, f 5 ) = (f 1 + f 3 f 5, f 1 f f 4 ). This map sujects onto J(σ 1 ) J(σ ) if and only if eithe J(013)

8 8 M. DIPASQUALE, C.A. FRANCISCO, J. MERMIN, AND J. SCHWEIG is geneated by x a, z c o J(01) is geneated by y b, (y x) d. This in tun happens if and only if eithe c + e a + 1 o b + d a + 1. By Theoem 3.9, (A G, m) is fee if and only if c + e a + 1 o b + d a + 1. Remak Let G be a gaph on n+1 vetices. As a consequence of Theoem 3.9, the Hilbet polynomial (and indeed Hilbet function) of D(A G, m), when (A G, m) is fee, is given by the Eule chaacteistic of R/J [G], namely HP (D(A G, m), d) = = dim (G) i=0 dim (G) i=0 ( 1) i HP (R/J [G] i, d) ( 1) i γ (G) i HP (S/J(γ), d). Assuming D(A G, m) is fee, geneated in degees 0, A 1,..., A k, we also have ( ) d + n 1 k ( ) d + n 1 Ai HP (D(A G, m), d) = +. n 1 n 1 Equating the leading coefficients of these two expessions yields k = n. Equating second coefficients yields the well-known expession A A n = m, whee m = ij m ij. Equating coefficients of d n 3 yields the equality of so-called second local and global mixed poducts, GMP () = LMP (), defined in [ATW07]. This gives some insight into how a bette undestanding of the homologies of R/J [G] will lead to moe pecise obstuctions to feeness. Indeed, the Hilbet polynomial takes no account of gaded dimensions that eventually vanish, while feeness may depend heavily on such infomation. It is this Atinian infomation that we now chaacteize. 3.. Feeness via syzygies. Fo the emainde of the pape, we specialize to the A 3 baid aangement. In this section we chaacteize fee multiplicities on A 3 in Theoem 3.16 as multiplicities fo which a cetain syzygy module is geneated locally. We label K 4 as in Figue 5. Just as in Remak.1, we choose vaiables x = x 1 x 0, y = x x 0, z = x 3 x 0. i=1 v 1 (z x) e zc v 0 x a y b (y x)d v 3 (z y) f v Figue 5. Complete gaph on fou vetices Lemma Fo any multiplicity m = (a, b, c, d, e, f), D(A K4, m) is fee if and only if H (J [K 4 ]) = 0.

9 FREE AND NON-FREE MULTIPLICITIES ON THE A 3 ARRANGEMENT 9 Poof. Since the clique complex (K 4 ) is a thee-dimensional simplex, it is contactible and H i (R[K 4 ]) = 0 except when i = 0. Fom the long exact sequence in homology associated to 0 J [K 4 ] R[K 4 ] R/J [K 4 ] 0, we conclude that H i (R/J [K 4 ]) = H i+1 (J [K 4 ]) fo i 1. It follows fom Theoem 3.9 that (A K4, m) is fee if and only if H i (J [K 4 ]) = 0 fo all i > 1. The complex J [K 4 ] has the fom 0 J(ij) J(ijk) J(013) 0. ij (K 4) 1 ijk (K 4) The final map is clealy sujective, so H 3 (J [K 4 ]) = 0. Hence (A K4, m) is fee if and only if H (J [K 4 ]) = 0. The following lemma gives a pesentation fo the homology module H (J [K 4 ]). Lemma Let K 4 have multiplicities m(τ) Z + fo each edge τ E(K 4 ). Endow the fomal symbols [e τ ] with degees m(τ). We define the module of locally geneated syzygies K Se τ as follows. Fo each σ (K 4 ), set τ E(K 4) K σ = { τ σ a τ [e τ ] : a τ α m(τ) τ = 0 and K = σ K σ. Also define the global syzygy module V V = τ E(K 4) } a τ [e τ ] : a τ ατ m(τ) = 0. Then K V and H (J [K 4 ]) = V/K as S-modules., τ E(K 4) S[e τ ] by Poof. The poof is vey simila to the poof of [SS97, Lemma 3.8]. Set up the following diagam with exact columns, whose fist ow is the complex J [K 4 ] τ E(K 4) J(τ) σ (K 4) J(σ) J(013) τ E(K 4) S[e τ ] σ (K 4) τ E(K 4) τ σ S[e τ,σ ] τ E(K 4) S[e τ ] 0 σ (K 4) K σ ι V 0 0

10 10 M. DIPASQUALE, C.A. FRANCISCO, J. MERMIN, AND J. SCHWEIG The middle ow is in fact exact. We ague this as follows. Given τ (K 4 ) 1, let τ be the sub-complex of (K 4 ) consisting of simplices which don t contain τ; τ is the union of two tiangles joined along the one edge which does not intesect τ. The middle ow splits as a diect sum of sub-complexes of the fom S[e τ ] S[e τ,σ1 ] S[e τ,σ ] S[e τ ], whee σ 1, σ ae the two tiangles which meet along τ. The (co)homology of each of these sub-complexes may be identified with the simplicial cohomology of (K 4 ) elative to τ, which vanishes in all dimensions. Now the long exact sequence in homology yields the isomophisms H 1 (J [K 4 ]) = ke(ι) and H (J [K 4 ]) = coke(ι). The image of K σ unde ι is pecisely σ (K 4) K, so we ae done. As a consequence of Theoem 3.9 and Lemma 3.15, the multiplicity m is fee if and only if the syzygy module of J(013) is locally geneated, as we summaize in the next theoem. Theoem The multiplicity m is fee on K 4 if and only if the syzygies on the ideal J(013) ae geneated by the syzygies on the fou sub-ideals J(01), J(013), J(03), J(13). With notation as in Figue 5, the multiplicity m = (a, b, c, d, e, f) is fee if and only if the syzygies on x a, y b, z c, (y x) d, (z x) e, (z y) f ae geneated by the syzygies on the fou sub-ideals x a, y b, (y x) d x a, z c, (z x) e y b, z c, (z y) d (z y) d, (z x) e, (y x) f. Coollay Let K 4 be labeled as in Figue 5. If J(013) is minimally geneated by thee of the six powes x a, y b, z c, (y x) d, (z x) e, (z y) f, and these thee coespond to all the edges adjacent to a single vetex, then D(A K4, m) is fee. Up to elabeling the vetices, this may be expessed by the thee simultaneous inequalities a + b d + 1, a + c e + 1, b + c f + 1. Remak Coollay 3.17 appeas in [ATW08, Coollay 5.1], whee the multiaangements (A 3, m) with these multiplicities ae additionally identified as inductively fee multi-aangements. Poof. In this case, the module syz(j(013)) is geneated by thee Koszul syzygies and thee elations of degee d, e, f, expessing (y + z) d, (x + z) e, (x y) f in tems of x a, y b, z c. Each of the modules syz(j(01)), syz(j(013)), syz(j(03)) contibutes a Koszul syzygy and one of the syzygies of degee d, e, f, espectively. Hence the syzygies on J(013) ae geneated by the syzygies on the fou sub-ideals. The esult follows fom Theoem Definition We call a vetex i satisfying the thee inequalities m jk m ij + m ik 1 of Coollay 3.17 a fee vetex; if one of 0, 1,, 3 is a fee vetex then we say that m has a fee vetex.

11 FREE AND NON-FREE MULTIPLICITIES ON THE A 3 ARRANGEMENT classification, pat I In this section we pove the classification of Theoem 1.1 fo multiplicities satisfying the inequalities m ij m ik + m jk + 1 fo all choices of i, j, k (giving a total of 1 iedundant inequalities). The eason fo imposing these inequalities is detailed at the beginning of Section 4.; biefly, these place estictions on the degees in which the syzygy modules syz(j(ijk)) ae geneated. The emaining multiplicities ae consideed in Section 5. Sections 4 and 5 taken togethe constitute the poof of Theoem Non-fee A 3 multiplicities via Hilbet function evaluation. By Theoem 3.16, we may establish that the multiplicity (A 3, m) is not fee by exhibiting a degee d in which the Hilbet functions of syz(j(ijk)) and syz(j(013)) diffe. In geneal it may be quite difficult to detemine these Hilbet functions; howeve, we ae able to obtain bounds. Thoughout, we adopt the convention that ( A B) = 0 if A < B. We begin by descibing a lowe bound on the global syzygies. Fom the exact sequence 0 syz(j(013)) i,j S( m ij ) S S/J(013) 0, we have HF (syz(j(013)) + HF (S) = i,j HF (S( m ij )) + HF (S/J(013)). Computing the Hilbet function of the module S/J(013) is difficult, so we settle fo the following inequality. Poposition 4.1. Fo all d, HF (syz(j(013), d) ( ) ( ) (d mij ) + d +. i,j Poof. Fom above, HF (syz(j(013)) i,j HF (S( m ij )) HF (S). Evaluating the ight-hand side at d gives the desied inequality. Remak 4.. The bound in Poposition 4.1 can be impoved (possibly made exact) by using invese systems [EI95] to evaluate dim syz(j(013)) d exactly via a fat point computation. This tanslates the ideal J(013) into a fat point ideal whose base locus is six points (coesponding to the six edges of K 4 ); these points ae the intesection points of fou geneic lines (coesponding to the fou tiangles of K 4 ). A complete classification of fat point ideals on six points, including thei Hilbet function and minimal fee esolution, appeas in [GH07]. Supisingly, the weake bound of Poposition 4.1 suffices fo the classification of fee multiplicities. Now we tun ou attention to the local syzygies. The Hilbet functions of the syzygies on the individual J(ijk) povide an uppe bound on the Hilbet function of the local syzygy module:

12 1 M. DIPASQUALE, C.A. FRANCISCO, J. MERMIN, AND J. SCHWEIG Poposition 4.3. HF J(ijk) HF (J(ijk)). i,j,k i,j,k The computation of the Hilbet functions of the individual local syzygy modules is moe technical and is done by Schenck [GS98], which we cite below in Lemma 4.5. Remak 4.4. Ou intuition fo Schenck s esult below is the following. Obseve that J(01) is isomophic to x a, y b, (y x) c. We study K[x, y]/ x a, y b, (y x) c, which is isomophic to ( )/ K[x, y] x a, y b (y x) c. Lemma 4.5 is equivalent to the statement that in this quotient ing, (y x) c is a Lefschetz element (i.e., multiplication by this element is eithe injective o sujective). The Hilbet function inceases as the degee deceases fom the socle degee (a+b ) to (a+b )/. On the othe hand, since (y x) c is a Lefschetz element, the Hilbet function of the ideal (y x) c in this quotient ing is 1 in degee c and inceases with the degee as long as possible. By the Hilbet-Buch Theoem, thee ae two minimal fist syzygies. Thei degees ae whee the ideal s Hilbet function would exceed that of the ing. Unfotunately, these degees depend on the paity of a, b, and c. The two mysteious quantities in the statement of Lemma 4.5, Ω ijk and a ijk, encode the paity cases simultaneously. The following lemma is an immediate consequence of [GS98, Theoem.7]. Lemma 4.5. Let J(ijk) = (x i x j ) mij, (x i x k ) m ik, (x j x k ) m jk S. Set mij + m jk + m ik 3 Ω ijk = + 1 and a ijk = m ij + m jk + m ik Ω ijk. Then, if (x i x j ) mij, (x i x k ) m ik, and (x j x k ) m jk ae a minimal geneating set, syz(j(ijk)) = S( Ω ijk 1) a ijk S( Ω ijk ) a ijk. Othewise, suppose without loss of geneality that m ij + m jk m ik + 1. Then syz(j(ijk)) = S( m ik ) S( m ij m jk ). Remak 4.6. We emak fo late use that if m ij m ik + m jk + 1 fo all i, j, k then syz(j(ijk)) = S( Ω ijk 1) a ijk S( Ω ijk ) a ijk, in othe wods, even if (x i x j ) mij, (x i x k ) m ik, and (x j x k ) m jk ae not quite a minimal geneating set fo J(ijk), the Betti numbes fo syz(j(ijk)) ae the same as if they wee. Poof. If (x i x j ) mij, (x i x k ) m ik, (x j x k ) m jk ae a minimal geneating set, then the minimal fee esolution of J(ijk) has the fom 0 S( Ω ijk 1) a ijk S( Ω ijk ) a ijk φ S( m ij ) S( m ik ) S( m jk ) by [GS98, Theoem.7]. Othewise, if m ij +m jk m ik +1 then J(ijk) is geneated by (x i x j ) mij, (x j x k ) m jk. So the syzygies on the geneatos (x i x j ) mij, (x i x k ) m ik, (x j x k ) m jk ae given by the Koszul syzygy on (x i x j ) mij, (x j x k ) m jk

13 FREE AND NON-FREE MULTIPLICITIES ON THE A 3 ARRANGEMENT 13 and a syzygy of degee m ik (expessing (x i x k ) m ik of (x i x j ) mij, (x j x k ) m jk ). See Remak 3.8. as a polynomial combination Remak 4.7. Since the module syz J(01) can be identified with the non-tivial deivations on the multi-aangement (A, m) = (A K3, m) (see Example 3.6), Lemma 4.5 also follows fom a esult of Wakamiko [Wak07] on the exponents of the multi-aangement (A, m). Combining the local and global bounds above, we poduce a citeion fo nonfeeness of the multi-aangement (A 3, m). Define the function LB(m, d) by [ ( ) ] ( ) d + mij d + LB(m, d) = HF (syz(j(ijk)), d) i,j i,j,k ( ) [ ( ) ] [ ] d + d + mij = 3 HF (S/J(ijk), d). i,j i,j,k The two diffeent expessions fo LB(m, d) ae the same; this is immediate fom the exact sequence 0 syz(j(ijk)) S( m ij ) S( m ik ) S( m jk ) S S/J(ijk) 0, which holds fo each i, j, k. Theoem 4.8. We have HF (syz J(013), d) HF syz J(ijk), d LB(m, d). i,j,k In paticula, if LB(m, d) > 0 fo any intege d 0, then (A 3, m) is not fee. Poof. The inequality HF (syz J(013), d) HF syz J(ijk), d LB(m, d) i,j,k follows immediately fom Popositions 4.1 and 4.3. By Theoem 3.16, m is a fee multiplicity on A 3 if and only if HF (syz J(013), d) HF syz J(ijk), d = 0 i,j,k fo all d Non-fee multiplicities via disciminant. The function LB(m, d) fom Theoem 4.8 is eventually polynomial in d. Denote the Hilbet polynomial by LB(m, d); this is quadatic with leading coefficient 3/. In this section we assume that all of the ideals J(ijk) ae close to minimally geneated by thei thee geneatos. Explicitly, we impose the inequalities m ij m ik + m jk + 1 fo all choices of i, j, k (giving a total of 1 iedundant inequalities). Following Remak 4.6 it is staightfowad to check that unde these assumptions, syz(j(ijk)) is geneated in degees Ω ijk + 1, Ω ijk + 1 if m ij + m ik + m jk is even and degees Ω ijk, Ω ijk + 1 if m ij + m ik + m jk is odd, whee the constants Ω ijk ae as in Lemma 4.5. Set I = {0, 1,, 3}. We have

14 14 M. DIPASQUALE, C.A. FRANCISCO, J. MERMIN, AND J. SCHWEIG LB(m, d) = {i,j} I ( ) ( ) d + mij d + (( ) d + 1 Ωijk + {i,j,k} I ( d + Ωijk )). Lemma 4.9. Let m = m ij. The polynomial LB(m, d) attains its maximum value at d max = 1 ( m 9). 6 Futhemoe, assume m does not have a fee vetex. Then LB(m, d) = LB(m, d) fo d d max. Poof. Using the second expession fo LB(m, d) (just pio to Theoem 4.8) and expanding the binomial coefficients as polynomials in d, we see that LB(m, d) is a quadatic polynomial Ad + Bd + C with A = 3/ B = 9/ + m C = 3 ij ( mij 1 ) ijk HP (S/J(ijk), d), whee HP (S/J(ijk), d) is the Hilbet polynomial of S/J(ijk) (since S/J(ijk) is zeo-dimensional as a scheme ove P, this is a constant). It follows immediately that LB(m, d) achieves its maximum at d max = ( m 9)/6. Fo the second claim, it suffices to show that (1) d max m ij fo all i, j, and () d max Ω ijk 1 fo all i, j, k. Fo the fist inequality, assume without loss of geneality that {i, j} = {0, 1}. We have m 3 (m 03 + m 13 ) (m 01 1) (m 0 + m 1 ) (m 01 1) m 01 m 01. Summing down this list of inequalities yields m 6m 01, so 1 d max = ( m 9) m = m Fo the second inequality, assume without loss of geneality that {i, j, k} = {0, 1, }. We have (m 01 + m 0 + m 1 ) (m 01 + m 0 + m 1 ) m 13 + m 03 m 01 1 m 03 + m 3 m 0 1 m 13 + m 3 m 1 1. Summing down this list we obtain m 3(m 01 + m 0 + m 1 ) 3. In fact, we will show that m 3(m 01 + m 0 + m 1 ). Assume to the contay that m < 3(m 01 + m 0 + m 1 ); then (m 03 + m 13 + m 3 ) < m 01 + m 0 + m 1. Reaanging yields (m 13 + m 03 m 01 ) + (m 03 + m 3 m 0 ) + (m 13 + m 3 m 1 ) < 0.

15 FREE AND NON-FREE MULTIPLICITIES ON THE A 3 ARRANGEMENT 15 Accoding to the displayed inequalities above, each of the thee paenthesized tems in the above sum is at least 1. Consequently, each of these tems must be at most 1, i.e. m 01 m 13 + m 03 1, m 0 m 03 + m 3 1, and m 1 m 13 + m 3 1. But then 3 is a fee vetex. So, assuming m does not have a fee vetex, we have m 3(m 01 +m 0 +m 1 ). Hence d max = 1 ( m 9) 6 m01 + m 0 + m = Ω Lemma Let D be the disciminant of the quadatic polynomial LB(m, d) in the vaiable d. (1) If D 9/4 > 0, then (A 3, m) is not fee. () If m 0 (mod 3) and D 1/4 > 0, then (A 3, m) is not fee. Poof. We examine when the polynomial LB(m, d) is positive at some intege d > 0. Fo this to happen, LB(m, d) must have two eal oots, say 1 and, and thee must be an intege stictly between them. Equivalently, thee must be an intege in the inteval Q = ( 1, ) = (d max 1 1, d max ). Fom the fom of d max given in Lemma 4.9, (1) If m 0 (mod 3) then d max = N + 1/ fo some intege N () If m 0 (mod 3) then d max = N ± 1/6 fo some intege N Fom the quadatic fomula and the fact that the leading coefficient of LB(m, d) is 3/, we have ( 1 ) = 4D /9. Hence if 4D /9 > 1, then Q contains an intege. Moeove, if m 0 (mod 3) and 4D /9 > 1/9, then Q also contains an intege. Now the esult follows fom Lemma 4.9 and Theoem 4.8. Remak In the following theoem, we pefomed the staightfowad but tedious computations with the compute algeba system Mathematica. Theoem 4.1. Let P (m) = (m 01 +m 3 m 0 m 13 ) +(m 0 +m 13 m 03 m 1 ) +(m 03 +m 1 m 01 m 3 ) and set m ijk = m ij + m jk + m ik. Assume that m ij m ik + m jk + 1 fo evey i, j, and k. Assume futhe that m does not have a fee vetex. If any of the conditions below ae satisfied, then m is not a fee multiplicity on A 3 = A K4. m 0 mod 3, none of the m ijk ae odd, and P (m) > 0 m 0 mod 3, two of the m ijk ae odd, and P (m) > 6 m 0 mod 3, fou of the m ijk ae odd, and P (m) > 1 m 0 mod 3 and none of the m ijk ae odd. m 0 mod 3, two of the m ijk ae odd, and P (m) > m 0 mod 3, fou of the m ijk ae odd, and P (m) > 8. Remak The polynomial P (m) of Theoem 4.1 is essentially an uppe bound on the diffeence between GMP () and LMP (), the second global and local mixed poducts intoduced in [ATW07]. Indeed, this theoem could be poved using these techniques. Poof of Theoem 4.1. Let D be the disciminant of LB(m, d). Fom the poof of Lemma 4.9, LB(m, d) = Ad + Bd + C with A = 3/

16 16 M. DIPASQUALE, C.A. FRANCISCO, J. MERMIN, AND J. SCHWEIG B = 9/ + m C = 3 ij ( mij 1 ) ijk HP (S/J(ijk), d). Hence D = B 4AC = 9 m + m 6 ij m ij + 6 ijk HP (S/J(ijk), d). The polynomial HP (S/J(ijk), d) is a constant, in fact, ( ) Ωijk + 1 ( ) Ωijk + 1 m st HP (S/J(ijk), d) =. {s,t} {i,j,k} Since the constant Ω ijk depends on the paity of m ijk = m ij + m ik + m jk, the disciminant D will also. A staightfowad computation now yields that (D 9/4) is equal to P (m) 3q, whee q is the numbe of m ijk that ae odd. Note that ijk m ijk = m, so q equals zeo, two, o fou. The dependence on the conguence class of m modulo thee follows fom Lemma In the case that m 0 mod 3 and none of the m ijk ae odd, (D 1/4) = P (m) + 4, which is always positive. Hence we always have non-feeness in this case. Definition Let n i Z 0 fo i = 0, 1,, 3 and ɛ ij { 1, 0, 1} fo 0 i < j 3. An ANN multiplicity on A 3 is a multiplicity of the fom m ij = n i + n j + ɛ ij. ANN multiplicities ae classified as fee o non-fee in [ANN09] (not just on A 3 but on any baid aangement). Poposition Let m be a multiplicity so that m ij m ik + m jk + 1 fo evey i, j, and k. Then m is a fee multiplicity fo A 3 if and only if m has a fee vetex o m is a fee ANN multiplicity. Poof. If m has a fee vetex then it is fee by Coollay We now show that if any of the conditions of Theoem 4.1 fail, then m is an ANN multiplicity. We will do this by explicitly constucting non-negative integes N 0, N 1, N, N 3 and ɛ ij { 1, 0, 1} so that m ij = N i + N j + ɛ ij fo 0 i < j 3. The main thing we have to be caeful about is the non-negativity of the N i. We intoduce some notation. Fo a vetex i of a tiangle ijk, set n i,ijk = (m ij + m ik m jk )/. Since we assume m jk m ij + m ik + 1 fo evey tiple i, j, k, it follows that n i,ijk 1/. Also, fo a diected fou-cycle ijst set c ijst = (m ij m js + m st m it )/. If all of the m ijk ae even, then evey expession n i,ijk is a non-negative intege. In this case, negating Theoem 4.1 means P (m) = 0; hence c ijst = 0 fo evey diected fou cycle, and the expessions n i,ijk ae independent of the tiangle chosen to contain i (fo instance, n 0,01 = n 0,03 = n 0,013 ). Set N 0 = n 0,01, N 1 = n 1,01, N = n,01, and N 3 = n 3,013. We have N i 0 and m ij = N i + N j fo all i, j, so m is an ANN multiplicity. Now suppose two of the m ijk ae odd, and P (m) 6. Suppose without loss of geneality that m 01 and m 03 ae even, while m 013 and m 13 ae odd. Set N 0 = n 0,01, N 1 = n 1,01, N = n,03, and N 3 = n 3,03. Note that, given ou assumptions, all the N i ae non-negative integes. N 0 + N 1 = n 0,01 + n 1,01 = m 01 N 0 + N = n 0,01 + n,03 = m 0 + c 013 N 0 + N 3 = n 0,01 + n 3,03 = m 03 + c 013 N 1 + N = n 1,01 + n,03 = m 1 + c 013 N 1 + N 3 = n 1,01 + n 3,03 = m 13 + c c 031 N + N 3 = n,03 + n 3,03 = m 3

17 FREE AND NON-FREE MULTIPLICITIES ON THE A 3 ARRANGEMENT 17 Note also that, unde ou assumptions, c 013 is an intege while c 013 and c 031 ae not. We also have c 013 +c 031 = c 013. Since P (m) 6, we have only the following possibilities: c 013 = 1, c 031 = 1/, c 013 = 1/ c 013 = 1, c 031 = 1/, c 013 = 1/ c 013 = 0, c 031 = c 013 = 1/ c 013 = 0, c 031 = c 013 = 1/. In any of the above situations, set ɛ 01 = ɛ 3 = 0, ɛ 0 = ɛ 03 = ɛ 1 = c 013, and ɛ 13 = c 013 c 031. By the above obsevations, we have shown m ij = N i +N j +ɛ ij is an ANN multiplicity. Finally, suppose all of the m ijk ae odd and P (m) 1. In fact, P (m) is the sum of squaes of thee integes which add to zeo, so inspection yields P (m) 8. Set Ñ0 = n 0,01, Ñ1 = n 1,013, Ñ = n,03, and Ñ3 = n 3,13. Note that, given ou assumptions, all the Ñi ae non-integes. We modify them shotly. We have Ñ 0 + Ñ1 = n 0,01 + n 1,013 = m 01 + c 013 Ñ 0 + Ñ = n 0,01 + n,03 = m 0 + c 013 Ñ 0 + Ñ3 = n 0,01 + n 3,13 = m 03 + c c 013 Ñ 1 + Ñ = n 1,013 + n,03 = m 1 + c c 013 Ñ 1 + Ñ3 = n 1,013 + n 3,13 = m 13 + c 013 Ñ + Ñ3 = n,03 + n 3,13 = m 3 + c 013. Unde ou assumptions, c 013, c 013, and c 031 ae all integes. We also have c c 031 = c 013. Since P (m) 8, at most two of c 013, c 013, and c 031 can be nonzeo, and all must have absolute value at most one. Fist assume c 013 = 0 and c 031 = ±1. We have Ñ 0 + Ñ1 = n 0,01 + n 1,013 = m 01 + c 013 Ñ 0 + Ñ = n 0,01 + n,03 = m 0 Ñ 0 + Ñ3 = n 0,01 + n 3,13 = m 03 + c 013 Ñ 1 + Ñ = n 1,013 + n,03 = m 1 + c 013 Ñ 1 + Ñ3 = n 1,013 + n 3,13 = m 13 Ñ + Ñ3 = n,03 + n 3,13 = m 3 + c 013. Since c and m ij 1 fo all i, j, at most one of the Ñi is equal to 1/. Without loss, assume Ñ0 1/ while N i 1/ fo i = 1,, 3. Now set N 0 = Ñ0, N 1 = Ñ1, N = Ñ, and N 3 = Ñ3. With these assumptions, we have N 0 + N 1 = n 0,01 + n 1,013 = m 01 + c 013 N 0 + N = n 0,01 + n, = m N 0 + N 3 = n 0,01 + n 3,13 = m 03 + c 013 N 1 + N = n 1,013 + n,03 = m 1 + c 013 N 1 + N 3 = n 1,013 + n 3,13 1 = m 13 1 N + N 3 = n,03 + n 3,13 = m 3 + c 013. So m is an ANN multiplicity with ɛ 01 = ɛ 03 = ɛ 1 = ɛ 3 = c 013, ɛ 0 = 1, and ɛ 13 = 1. The case c 013 = 0 is symmetic to the above case. We now conside the case c 031 = 0, which implies c 013 = c 013. If c 013 = 0 as well, then we again have at most one of Ñ i equal to 1/, and we ague that m is an ANN multiplicity in the same way as above.

18 18 M. DIPASQUALE, C.A. FRANCISCO, J. MERMIN, AND J. SCHWEIG Now suppose that c 031 = 0 and c 013 = c 013 = 1. Then Ñ 0 + Ñ1 = n 0,01 + n 1,013 = m Ñ 0 + Ñ = n 0,01 + n,03 = m Ñ 0 + Ñ3 = n 0,01 + n 3,13 = m 03 + Ñ 1 + Ñ = n 1,013 + n,03 = m 1 + Ñ 1 + Ñ3 = n 1,013 + n 3,13 = m Ñ + Ñ3 = n,03 + n 3,13 = m In this case it is also clea that at most one of Ñ i can equal 1/. If all Ñ i ae at least 1/, then we can take N i = Ñi fo i = 0, 1,, 3. Then we will clealy have an ANN multiplicity. Suppose then that one of the Ñi is equal to 1/. Without loss of geneality we can assume that Ñ0 = 1/. Using the thid listed equation above, Ñ 3 7/. In this case we can set N 0 = Ñ0 = 0, N 1 = Ñ1, N = Ñ, and N 3 = Ñ3 1, giving an ANN multiplicity. Finally, suppose that c 031 = 0 and c 013 = c 013 = 1. Then Ñ 0 + Ñ1 = n 0,01 + n 1,013 = m 01 1 Ñ 0 + Ñ = n 0,01 + n,03 = m 0 1 Ñ 0 + Ñ3 = n 0,01 + n 3,13 = m 03 Ñ 1 + Ñ = n 1,013 + n,03 = m 1 Ñ 1 + Ñ3 = n 1,013 + n 3,13 = m 13 1 Ñ + Ñ3 = n,03 + n 3,13 = m 3 1. Set N i = Ñi fo i = 0, 1,, 3. Then N i 0 fo i = 0, 1,, 3 and we have an ANN multiplicity. 5. classification, Pat II In this section we complete the classification of fee multiplicities on A 3 given in Theoem 1.1. Ou stategy is to show that, if we assume m has no fee vetex and that the syzygies of J(013) ae locally geneated as equied by Theoem 3.16, then we ae foced to have the twelve inequalities m ij m ik + m jk + 1 fo evey tiple i, j, k. Then Poposition 4.15 guaantees that such a multiplicity is fee if and only if it is a fee ANN multiplicity. We intoduce some notation fo studying the local syzygies. Notation 5.1. Label the exponents with the lettes a though f as in Figue 1, and efe to the foms as A = (x 1 x 0 ) a, and so on. The local ideals J(01), J(013), J(03), and J(13) then have (not necessaily minimal) geneating sets {A, B, D}, {A, C, E}, {B, C, F }, and {D, E, F }. Notation 5.. Conside the fee S-module of ank six with basis [A], [B],..., [F ]. A syzygy on J(013) is an expession of the fom g a [A] + g b [B] + g c [C] + g d [D] + g e [E] + g f [F ] satisfying g a A + g b B + g c C + g d D + g e E + g f F = 0. Its suppot is the set of geneatos with nonzeo coefficient; fo example, the Koszul syzygy A[B] B[A] has suppot {A, B}. We say that a syzygy is local if its suppot is a subset of {A, B, D}, {A, C, E}, {B, C, F }, o {D, E, F }, and locally geneated if it is a linea combination of local syzygies.

19 FREE AND NON-FREE MULTIPLICITIES ON THE A 3 ARRANGEMENT 19 Notation 5.3. We intoduce notation, and an abuse theeof, to descibe the syzygies on the local ideal J(01) = A, B, D. We extend this notation to the othe tiangles in the obvious way. We denote the Koszul syzygy A[B] B[A] by K ab ; it has degee a + b. Similaly, the Koszul syzygies K ad and K bd have degees a + d and b + d espectively. The suppot of the Koszul syzygy K ab is {A, B}. Thee ae also syzygies with suppot {A, B, D}; fom Lemma 4.5 these have degee as low as a+b+d 1 (when {A, B, D} is a minimal geneating set) and as low as d if D is not a minimal geneato (with obvious adjustments fo symmety). Since many of ou aguments below concen only the suppots of the syzygies, we abuse notation and efe to any syzygy with suppot {A, B} by the name K ab. (Thus, while K ab may not efe to the Koszul syzygy, it does efe to an S-linea multiple, so all elevant intuition about Koszul syzygies continues to wok.) Finally, S abd will efe to any syzygy suppoted on a subset of {A, B, D}. Without loss of geneality, let K be have the least degee among the non-local Koszul syzygies K af, K be, K cd. We will show that if m is a fee multiplicity with no fee vetex and K be is locally geneated (as it must be by Theoem 3.16), then m is a fee ANN multiplicity. To that end we make the following assumptions fo the emainde of the section. Assumptions 5.4. (1) Thee is no fee vetex. () b + e min{a + f, c + d} (3) K be is locally geneated. That is, we may wite ( ) K be = S abd + S ace + S bcf + S def. Lemma 5.5. Given Assumptions 5.4 and efeing to Equation ( ), If S def is not suppoted on E then e a + c 1 If S ace is not suppoted on E then e d + f 1 If S abd is not suppoted on B then b c + f 1 If S bcf is not suppoted on B then b a + d 1. Poof. We pove the fist statement. The emaining statements ae poved in the same way. Fixing coodinates, we may wite A = x a, B = y b, C = z c, E = (x z) e. Obseve S ace = g a [A] + g c [C] + g e [E], whee g a, g b, g c S. On the one hand, g e E = (g a A + g c C), so g e ( A, C : E). On the othe hand, since we assumed S def is not suppoted on E, no othe tems in Equation ( ) ae suppoted on [E], so g e = B. In paticula, B ( A, C : E). In othe wods, y b ( x a, z c : (x z) e ), so we conclude that ( x a, z c : (x z) e ) = 1. Consequently, E A, C, which happens if and only if e a + c 1. Lemma 5.6. Given Assumptions 5.4 and efeing to Equation ( ), S ace and S def must both be suppoted on the edge E. Likewise S abd and S bcf must both be suppoted on B. Poof. In light of Lemma 5.5, it suffices to show that we have the fou stict inequalities e < a + c 1, e < d + f 1, b < c + f 1, and b < a + d 1. We show the inequality e < a + c 1; the est follow by symmety. Suppose to the contay

20 0 M. DIPASQUALE, C.A. FRANCISCO, J. MERMIN, AND J. SCHWEIG that e a + c 1. Then, since b + e min{a + f, c + d}, a + f b + e b + a + c 1 c + d b + e b + a + c 1. Consequently we have f b+c 1 and d b+a 1. Since we assumed e a+c 1, we conclude that vetex 0 is a fee vetex, violating Assumption 5.4.(1). Notation 5.7. We say that an edge is in the suppot of the local expession ( ) fo K be if it is in the suppot of one of the summands. Lemma 5.8. In the local expession K be = S abd + S ace + S bcf + S def, each summand must be suppoted on thee edges. Poof. By Lemma 5.6, we aleady know that each summand is suppoted on eithe E o B. Next we claim that the local expession fo K be must be suppoted on at least thee of the edges A, C, D, and F. Suppose to the contay that two of these edges ae absent fom the suppot. Up to symmety thee ae two possibilities: eithe the two edges ae adjacent (A and C) o the two edges ae opposite (A and F ). In the fist case, we have S ace = 0, contadicting Lemma 5.6. If the local expession is suppoted on A and F, then this foces K be = K bd + K ce + K bc + K de, which is impossible due to degee consideations, as we now explain. Looking at coefficients on [C] yields (pe +qb)[c] = 0, so pe +qb = 0. Since B and E have no common facto, deg(pe) b + e, so deg(k ce ) b + c + e > b + e, a contadiction. (If p = q = 0 then S ace = S def = 0, again contadicting Lemma 5.6.) Now suppose that the local expession fo K be is suppoted on all but one of the edges A, C, D, and F, without loss of geneality the edge A. Then we have the equation below. K be = ( E[B] B[E] ) = g abd ( D[B] B[D] ) + g ace ( E[C] C[E] ) + g bcf ( j b [B] +j c [C] j f [F ] ) + g def ( h d [D] +h e [E] +h f [F ] ). Equating coefficients on [B] and inspecting degees yields d e, while equating coefficients on [E] yields c b. Since b + e c + d, this implies d = e and c = b. Since c+e = b+e, we conclude that g ace is a scala, so, looking at the coefficients on [C], we conclude that (up to scala) g bcf j c = E. Thus S bcf is equivalent (up to scala) to EC = g bcf j b B + g bcf j f F, and we conclude EC B, F. But B, F is a pimay ideal and E n is not in B, F fo any n (since B, E, F fom a egula sequence), so C B, F, i.e. b + f c + 1. Since b = c, this implies f = 1. But then we have the multiplicity (a, b, b, d, d, 1) and b + e = b + d a + f = a + 1, so we have a fee vetex (in fact, vetices and 3 ae both fee), a contadiction. Now we show that each of the local syzygies is suppoted on all thee of its edges. It is enough to do this fo S abd = α abd [A] + β abd [B] + δ abd [D]. We aleady know fom Lemma 5.6 that S ace is suppoted on B (i.e. β abd 0). It suffices to show that S abd is suppoted on A (the agument fo suppot on D is the same). Adding coefficients on [A] in Equation ( ) yields α abd + α ace = 0. If S abd is not suppoted

21 FREE AND NON-FREE MULTIPLICITIES ON THE A 3 ARRANGEMENT 1 on A, then α abd = 0, so α ace = 0 as well. Then the local expession ( ) is not suppoted on A, a contadiction. We ae now eady to complete the poof of Theoem 1.1. Poposition 5.9. If m is a fee multiplicity without a fee vetex, then m ij m ik + m jk + 1 fo evey tiple i, j, k. Poof. Theoem 3.16 guaantees Assumption 5.4.(3), so we may take all of Assumptions 5.4 without loss. By the poof of Lemma 5.6, we aleady have (sticte vesions of) the fou inequalities b c + f + 1, b a + d + 1, e a + c + 1, and e d + f + 1. Hence we need to establish the eight emaining inequalities with a, c, d, and f on the left-hand side. We demonstate the inequality a b + d + 1. By symmety, the emaining seven inequalities ae established in pecisely the same way. Since b + e c + d, we have b + e (b + c + d + e)/. By Lemma 5.8, S ace is suppoted on [A], [C], and [E]. It follows that the degee of S ace is at least (a + c + e 1)/ by Lemma 4.5. Since S ace appeas in the expession fo K be, we have a + c + e 1 b + e b + c + d + e. Simplifying yields a b + d + 1, as desied. Poof of Theoem 1.1. Suppose m is a fee multiplicity without a fee vetex. By Poposition 5.9, m ij m ik + m jk + 1 fo evey tiple i, j, k. By Poposition 4.15, m must be a fee ANN multiplicity. Remak A defomation of the A 3 aangement (technically, the cone ove a defomation of the A 3 aangement) is a cental hypeplane aangement of the fom x = α 1 w,..., α a w y = β 1 w,..., β b w z = κ 1 w,..., κ c w y x = δ 1 w,..., δ d w z x = ɛ 1 w,..., ɛ e w y z = φ 1 w,..., φ f w w = 0, whee α i, β i, κ i, δ i, ɛ i, φ i ae all elements of the gound field K. Aangements of this type wee fist investigated systematically by Stanley [Sta96] and have since been the subject of many eseach papes. Ou esults may be used to show that feeness of a defomation of the A 3 aangement can be detected just fom its intesection lattice. This is eadily deduced fom geneal chaacteizations of feeness due to Yoshinaga [Yos04] and Abe-Yoshinaga [AY13]. Integal to both of these chaacteizations is the feeness of the multi-aangement obtained fom esticting the aangement to a chosen hypeplane, whee the multiplicity assigned to each hypeplane H in the estiction counts the numbe of hypeplanes that estict to H. In the case of a defomation of A 3, esticting to the hypeplane w = 0 clealy esults in the multi-aangement (A 3, (a, b, c, d, e, f)); feeness of this multi-aangement is detemined fom Theoem 1.1.

22 M. DIPASQUALE, C.A. FRANCISCO, J. MERMIN, AND J. SCHWEIG 6. Abe-Nuida-Numata multiplicities In this section we elate ou esults moe closely to the classification of ANN multiplicities by Abe-Nuida-Numata in [ANN09]. We fist state thei classification pecisely fo the A 3 aangement. We then show that the non-fee multiplicities in thei classification follow fom Theoem 4.1 and Poposition 5.9. Finally, we illustate how the fee multiplicities in thei classification may be used to povide the minimal fee esolution of the ideal J(013) geneated by powes of linea foms. We intoduce the notation fom [ANN09]. Let G be a signed gaph on fou vetices. That is, each edge of G is assigned eithe a + o a, and so the edge set E G decomposes as a disjoint union E G = E + G E G. Define 1 {i, j} E + G m G (ij) = 1 {i, j} E G 0 othewise. The gaph G is signed-eliminable with signed-elimination odeing ν : V (G) {0, 1,, 3} if ν is bijective, and, fo evey thee vetices v i, v j, v k V (G) with ν(v i ), ν(v j ) < ν(v k ), the induced subgaph G vi,v j,v k satisfies the following conditions. Fo σ {+, 1}, if {v i, v k } and {v j, v k } ae edges in EG σ then {v i, v j } EG σ Fo σ {+, 1}, if {v k, v i } EG σ and {v i, v j } E σ G then {v k, v j } E G Fo a signed-eliminable gaph G with signed elimination odeing ν, v V G and i {0, 1,, 3}, define the degee deg i (v) by deg i (v) := deg(v, V G, E + G ν 1 {1,...,i}) deg(v, V G, E G ν 1 {1,...,i}), whee deg(w, V H, E H ) is the degee of the vetex w in the gaph (V H, E H ) and (V G, E σ G S) with espect to S V G is the induced subgaph of G whose edge set is {{v i, v j } E σ G v i, v j S}. Futhemoe set deg i = deg i (ν 1 (i)) fo i = 0, 1,, 3. All signed-eliminable gaphs on fou vetices ae listed (with an elimination odeing) in [ANN09, Example.1], along with those which ae not signed-eliminable. Fo use in the poof of Coollay 6., we also list those gaphs which ae not signedeliminable in Table 1. The popety of being signed-eliminable is peseved unde intechanging + and. Consequently, we list these gaphs in Table 1 up to automophism with the convention that a single edge takes one of the signs +,, while a double edge takes the othe sign. Theoem 6.1. [ANN09, Theoem 0.3] Let k, n 0, n 1, n, and n 3 be nonnegative integes, and G be a signed gaph on fou vetices. Define the multiplicity m on the baid aangement A 3 by m ij = k+n i +n j +m G (ij). Set N = 4k+n 0 +n 1 +n +n 3. Assume one of the thee conditions: (1) k > 0 () E G = (3) E + G = and m ij > 0 fo evey {i, j} E K4. Then (A 3, m) is fee with exponents (0, N + deg, N + deg 3, N + deg 3 ) if and only if G is signed-eliminable. We fist show how we can ecove the non-fee ANN multiplicities on A 3 using Theoem 4.1.

23 FREE AND NON-FREE MULTIPLICITIES ON THE A 3 ARRANGEMENT 3 Table 1. Gaphs on fou vetices which ae not signed-eliminable Coollay 6.. Let k, n 0, n 1, n, and n 3 be non-negative integes, let G be a signed gaph on K 4, and let m be the ANN multiplicity m ij = k + n i + n j + m G (ij). If one of the two following conditions is satisfied, then m is not a fee multiplicity. (1) One o moe of the inequalities {m ij + m ik + 1 m jk 0 i < j < k 3} fails and m does not have a fee vetex. () All of the inequalities {m ij + m ik + 1 m jk 0 i < j < k 3} ae satisfied and G is not signed-eliminable. Poof. If the ANN multiplicity fails one o moe of the inequalities {m ij +m ik +1 m jk 0 i < j < k 3}, then it is fee if and only if it has a fee vetex by Poposition 4.15, completing the poof of (1). We now assume the inequalities m ij + m ik + 1 m jk on all tiples 0 i < j < k 3. We apply Theoem 4.1. It is evident that P (m ij ) = P (k + n i + n j + m G (ij)) = P (m G (ij)). Hence it is enough to show that P (m G (ij)) satisfies one of the inequalities of Theoem 4.1 if G is not signed-eliminable. This can be veified on a case-by-case basis; going acoss Table 1 fom left to ight and top to bottom: Two of m ijk odd, P (m G (ij)) = 14 > 6 None of m ijk odd, P (m G (ij)) = 8 > 0 None of m ijk odd, P (m G (ij)) = 8 > 0 None of m ijk odd, P (m G (ij)) = 8 > 0 Two of m ijk odd, P (m G (ij)) = 14 > 6 Two of m ijk odd, P (m G (ij)) = 18 > 6 None of m ijk odd, P (m G (ij)) = 4 > 0 Two of m ijk odd, P (m G (ij)) = 18 > 6 Two of m ijk odd, P (m G (ij)) = 14 > 6 Two of m ijk odd, P (m G (ij)) = 6 > 6 All of m ijk odd, P (m G (ij)) = 4 > 1 All of m ijk odd, P (m G (ij)) = 3 > 1 We conclude by emaking on how to use fee ANN multiplicities to constuct the minimal fee esolution of the ideal S/J(013). Coollay 6.3. The multi-aangement (A 3, m) is fee if and only if D(A 3, m) is a thid syzygy module of S/J(013) (in a non-minimal esolution). Poof. If D(A 3, m) is a thid syzygy module, it is fee by the Hilbet Syzygy Theoem. On the othe hand, suppose D(A 3, m) is fee. Let K = σ K σ be as in Lemma 3.15 and the inclusion ι : σ K σ V be as in the diagam in the poof of Lemma Conside the chain complex 0 ke(ι) σ (K 4) K σ ι τ E(K 4) Se τ S S/J(013) 0

24 4 M. DIPASQUALE, C.A. FRANCISCO, J. MERMIN, AND J. SCHWEIG Since D(A 3, m) is fee, the above complex is exact by Theoem The modules K σ, being syzygy modules of codimension two ideals, ae fee modules. The long exact sequence in homology applied to the diagam in the poof of Lemma 3.15 yields that H 1 (J [K 4 ]) = ke(ι). Since D(A 3, m) = H 0 (R/J [K 4 ]) = S H 1 (J [K 4 ]), D(A 3, m) is a (non-minimal) thid syzygy of S/J(013). Remak 6.4. While the minimal fee esolution of an ideal in two vaiables geneated by powes of linea foms is known (see [GS98]), thee is elatively little known about minimal fee esolutions of ideals geneated by powes of linea foms in thee vaiables. See [Sch04, Conjectue 6.3] fo a conjectue on the minimal fee esolution fo an ideal geneated by powes of seven linea foms in thee vaiables. Using Coollay 6.3, we can use the esult of Abe-Nuida-Numata to constuct the minimal fee esolution of J(013) wheneve m is a fee ANN multiplicity. Coollay 6.5. Let G be a signed-eliminable gaph on fou vetices with signedelimination odeing ν. Let k, n 0, n 1, n, n 3 be nonnegative integes and m be the multiplicity on A 3 with m ij = k+n i +n j +m G (ij). Also set N = 4k+(n 0 +n 1 +n + n 3 ), and let Ω ijk be as in Lemma 4.5. Then the ideal J(013) = (x i x j ) mij 0 i < j 3 has fee esolution: 0 3 i=1 S( N deg i ) i,j,k ( S( Ωijk ) a ijk S( Ω ijk 1) a ijk) S( m ij ) J(013) Futhemoe, if none of the six geneatos ae edundant, this esolution is minimal. We descibe thee special cases of Coollay 6.5. If m is constant with m ij = k, then 0 S( 4k) 3 S( 3k) 8 S( k) 6 S is a minimal fee esolution fo S/J(013). If m is constant with m ij = k + 1, then to use Coollay 6.5 we take G to be the complete gaph on fou vetices with all edges signed positively. Then deg = 1, deg =, and deg 3 = 3. Hence 0 S( 4k 1) S( 4k ) S( 4k 3) S( 3k 1) 4 S( 3k ) 4 S( k 1) 6 S is a minimal fee esolution fo S/J(013). Finally, suppose that m ij = n i + n j fo positive integes n 0, n 1, n, n 3. Then i,j 0 S( n i ) 3 ijk S( n i n j n k ) i,j S( n i n j ) S is a minimal fee esolution fo S/J(013). Acknowledgments: We ae especially gateful to Takuo Abe fo helpful comments and fo pointing out the application to defomations of A 3, which we have included in Remak We ae futhe gateful to Hal Schenck, Alexanda Seceleanu, Jianyun Shan, and Max Wakefield fo feedback on ealie dafts of the pape. We used Macaulay [GS] and Mathematica [Res16] in ou computations. Some of the computing fo this poject was pefomed at the OSU High Pefomance Computing Cente at Oklahoma State Univesity suppoted in pat though the National Science Foundation gant ACI The wok in this pape was patially suppoted by gants fom the Simons Foundation (#19914 to Chistophe Fancisco and #0115 to Jeffey Memin).

25 FREE AND NON-FREE MULTIPLICITIES ON THE A 3 ARRANGEMENT 5 Refeences [Abe07] Takuo Abe. Fee and non-fee multiplicity on the deleted A 3 aangement. Poc. Japan Acad. Se. A Math. Sci., 83(7):99 103, 007. [ANN09] Takuo Abe, Koji Nuida, and Yasuhide Numata. Signed-eliminable gaphs and fee multiplicities on the baid aangement. J. Lond. Math. Soc. (), 80(1):11 134, 009. [ATW07] Takuo Abe, Hioaki Teao, and Max Wakefield. The chaacteistic polynomial of a multiaangement. Adv. Math., 15():85 838, 007. [ATW08] Takuo Abe, Hioaki Teao, and Max Wakefield. The Eule multiplicity and additiondeletion theoems fo multiaangements. J. Lond. Math. Soc. (), 77(): , 008. [AY09] Takuo Abe and Masahiko Yoshinaga. Coxete multiaangements with quasi-constant multiplicities. J. Algeba, 3(8): , 009. [AY13] Takuo Abe and Masahiko Yoshinaga. Fee aangements and coefficients of chaacteistic polynomials. Math. Z., 75(3-4): , 013. [DiP16] M. DiPasquale. Genealized Splines and Gaphic Aangements. J. Algebaic Combin., 016. doi: /s [EI95] J. Emsalem and A. Iaobino. Invese system of a symbolic powe. I. J. Algeba, 174(3): , [GH07] Elena Guado and Bian Haboune. Resolutions of ideals of any six fat points in P. J. Algeba, 318(): , 007. [GS] Daniel R. Gayson and Michael E. Stillman. Macaulay, a softwae system fo eseach in algebaic geomety. Available at [GS98] Anthony V. Geamita and Heny K. Schenck. Fat points, invese systems, and piecewise polynomial functions. J. Algeba, 04(1):116 18, [OT9] Pete Olik and Hioaki Teao. Aangements of hypeplanes, volume 300 of Gundlehen de Mathematischen Wissenschaften [Fundamental Pinciples of Mathematical Sciences]. Spinge-Velag, Belin, 199. [Res16] Wolfam Reseach. Mathematica, Vesion 10.4, 016. [Sai75] Kyoji Saito. On the unifomization of complements of disciminant loci. In Confeence Notes. Ame. Math. Soc. Summe Institute, Williamstown, [Sai80] Kyoji Saito. Theoy of logaithmic diffeential foms and logaithmic vecto fields. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 7():65 91, [Sch97] Hal Schenck. A spectal sequence fo splines. Adv. in Appl. Math., 19(): , [Sch04] Heny K. Schenck. Linea systems on a special ational suface. Math. Res. Lett., 11(5-6): , 004. [Sch14] Hal Schenck. Splines on the Alfeld split of a simplex and type A oot systems. J. Appox. Theoy, 18:1 6, 014. [SS97] Hal Schenck and Mike Stillman. Local cohomology of bivaiate splines. J. Pue Appl. Algeba, 117/118: , Algoithms fo algeba (Eindhoven, 1996). [ST98] Louis Solomon and Hioaki Teao. The double Coxete aangement. Comment. Math. Helv., 73():37 58, [Sta96] Richad P. Stanley. Hypeplane aangements, inteval odes, and tees. Poc. Nat. Acad. Sci. U.S.A., 93(6):60 65, [Te0] Hioaki Teao. Multideivations of Coxete aangements. Invent. Math., 148(3): , 00. [Wak07] Atsushi Wakamiko. On the exponents of -multiaangements. Tokyo J. Math., 30(1):99 116, 007. [Yos0] Masahiko Yoshinaga. The pimitive deivation and feeness of multi-coxete aangements. Poc. Japan Acad. Se. A Math. Sci., 78(7): , 00. [Yos04] Masahiko Yoshinaga. Chaacteization of a fee aangement and conjectue of Edelman and Reine. Invent. Math., 157(): , 004. [Zie89] Günte M. Ziegle. Multiaangements of hypeplanes and thei feeness. In Singulaities (Iowa City, IA, 1986), volume 90 of Contemp. Math., pages Ame. Math. Soc., Povidence, RI, 1989.

26 6 M. DIPASQUALE, C.A. FRANCISCO, J. MERMIN, AND J. SCHWEIG Appendix A. Two-Valued Families In this appendix we illustate pictoially the classification of Theoem 1.1 fo two-valued multiplicities on A 3. Given two positive integes and s, we assume m ij = o m ij = s fo all i, j. In Table, the labeling of K 4 in the left column shows the assignment of multiplicities and the gaph on the ight shows which pais (, s) coespond to fee multiplicities (the obvious pattens continue). The hollow dots epesent fee multiplicities, while the solid dots epesent non-fee multiplicities. If pesent, the vetical line of fee multiplicities along = 1 coesponds to multiplicities with a fee vetex. Fee multiplicities clusteed aound the diagonal coespond to fee ANN multiplicities. s s s s s s Fee fo all, s 1

27 FREE AND NON-FREE MULTIPLICITIES ON THE A 3 ARRANGEMENT 7 s s s s s Table. Fee (hollow) and non-fee (solid) two-valued multiplicities on A 3 Michael DiPasquale, Depatment of Mathematics, Oklahoma State Univesity, Stillwate, OK , USA addess: mdipasq@okstate.edu URL: Chistophe A. Fancisco, Depatment of Mathematics, Oklahoma State Univesity, Stillwate, OK , USA addess: chis.fancisco@okstate.edu URL: Jeffey Memin, Depatment of Mathematics, Oklahoma State Univesity, Stillwate, OK , USA addess: memin@math.okstate.edu URL: Jay Schweig, Depatment of Mathematics, Oklahoma State Univesity, Stillwate, OK , USA addess: jay.schweig@okstate.edu URL: