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LINEAR ORBITS OF SMOOTH PLANE CURVES

arXiv:alg-geom/9206001v1 2 Jun 1992LINEAR ORBITS OF SMOOTH PLANE CURVESPaolo AluffiCarel Faber§1. IntroductionConsider a general codimension-8 linear subspace of the P14 parametrizing planequartic curves.

There is a generically finite dominant rational map from this P6to the moduli space of curves of genus 3; what is the degree of this map?To approach this kind of questions, we embark in this paper on the study of thenatural action of PGL(3), the group of automorphisms of P2, on the projectivespace PN parametrizing plane curves of degree d (thus N = d(d + 3)/2). We areconcerned here with the orbits of (points corresponding to) smooth plane curvesC of degree d ≥3.

These orbits OC are 8-dimensional quasi-projective (in fact,affine) varieties. Their closures OC (in PN) are 8-dimensional projective varieties,and one easily understands that the answer to the above question is nothingbut the degree of OC for the general plane quartic curve C. In §2 we explicitlyconstruct a resolution of these varieties, which we use in §3 to compute their degree(for every smooth plane curve C of degree d ≥3).

In fact the construction givesmore naturally the so-called predegree of the orbit closure OC: that is, the degreeof OC multiplied by the order of the PGL(3)-stabilizer of C. It turns out that, fora smooth curve C of degree d, the predegree depends only on d and the nature ofthe flexes of C. As is illustrated in §3.6, this has nice consequences related to theautomorphism groups of smooth plane curves.We now describe the contents of this paper more precisely. Associated witheach plane curve C is a natural map PGL(3) →PN with image OC; we viewthis as a rational map φC from the P8 of 3 × 3 matrices to PN.

In §2 our objectis to resolve this map by a sequence of blow-ups over P8, in fact constructing anon-singular compactification of PGL(3) that dominates OC. For a smooth C,the base locus of φC is a subvariety of P8 isomorphic to P2 × C, thus smooth;after blowing up this support, we find that the support of the base locus of theinduced rational map from the blow-up to PN is again smooth: and we choose itas the center of a second blow-up.

Just continuing this process (which requiresa fair amount of bookkeeping) gives a good resolution of the map. We find thatthe number of blow-ups needed equals the maximum order of contact of C with aline: so, for example, three blow-ups suffice for the general curve.Having resolved the map φC, we compute in §3 the predegree of the closure ofthe orbit of C as the 8-fold self-intersection of the pull-back of the class of a ‘point-condition’, i.e., a hyperplane in PN parametrizing the curves passing through agiven point of P2.

The main tool is an intersection formula for blow-ups from[Aluffi1]; to apply this formula, we extract from the geometry of the blow-ups1

2PAOLO ALUFFI CAREL FABERdetailed in §2 the relevant intersection-theoretic information, and particularly thenormal bundles and intersection rings of the centers of the various blow-ups. Thisleads to explicit formulas for the predegree of OC in terms of the degree of C andof four numbers encoding the number and type of the flexes of C. For example,the answer to the question posed in the beginning (that is, the degree for a generalquartic) is ,(d = 4 in the Corollary in §3.5).Besides the applications to automorphism groups of plane curves already al-luded to, and more examples given in §3.6 (e.g., we compute the degree of thetrisecant variety to the d-uple embedding of P2 in PN), the computation of thedegree of OC also has some enumerative significance: it gives the number of trans-lates of C that pass through 8 points in general position.

On a more global level,the degree of the orbit closure of a general plane curve of degree d equals thedegree of the natural map from a general codimension-8 linear subspace of PN tothe moduli space of smooth plane curves of degree d. A study of the boundariesof orbits and of 1-dimensional families of orbits reveals where this map is proper:these matters will be treated in a sequel to this paper. Also, we hope to be ableto unify some scattered results we have concerning orbits of singular curves; andwe plan to study the singular locus of orbit closures.Excellent practice to become familiar with the techniques of the paper is toapply them to the easier case of the action of the group PGL(2) on the spaces Pdparametrizing d-tuples of points on a line.

Only one blow-up of the P3 of 2 × 2matrices is needed in this case, and one finds the following: if the d-tuple consistsof points p1, . .

. , ps, with multiplicities mi (so that Psi=1 mi = d), and one putsm(2) = P m2i , m(3) = P m3i , then the predegree of its orbit-closure equalsd3 −3dm(2) + 2m(3),so it depends only on d and on m(2), m(3) (one should compare this result withthe degree of the orbit-closure of a smooth plane curve as computed in TheoremIII(B)).

Details of this computation, together with a discussion of the boundaryand of 1-dimensional families of orbits, and multiplicity results, can be foundin [Aluffi-Faber]. We should point out that in this case the degree can also becomputed by using simple combinatorics.Finally, to attract the attention of people working in representation theory, weremark that we deal here with the orbits of general vectors in one of the standardrepresentations of one of the classical algebraic groups.

Can these questions beapproached in a more general context? From this point of view, a whole lot ofwork remains to be done.Acknowledgements.

We thank the University of Chicago, Oklahoma StateUniversity, the Universit¨at Erlangen-N¨urnberg and the Max-Planck-Institut f¨urMathematik for hospitality at various stages of this work. Discussions with manyof our colleagues at these and other institutions were helpful.

We also acknowledgepartial support by the N.S.F. Finally, we are grateful to F. Schubert for writinghis wonderful Fantasia in F minor for piano duet.§2.

A blow-up constructionIn this section we construct a smooth projective variety surjecting onto the

LINEAR ORBITS OF SMOOTH PLANE CURVES3orbit closure OC of a smooth plane curve C ∈PN = Pd(d+3)2, where d ≥3. As wewill see, the construction depends essentially on the number and type of flexes ofC.Fix coordinates (x0 : x1 : x2) of P2, and assume the degree-d curve C hasequationF(x0, x1, x2) = 0.Consider the projective space P8 = PHom(C3, C3) of (homogeneous) 3×3 matricesα = (αij)i,j=0,1,2.

So P8 is a compactification of PGL(3) = {α ∈P8 : det α ̸= 0}.To ease notations, in this section we will refer to a point in P8 and to any 3 × 3matrix representing it by the same term; in the same vein, for α ∈P8 we will call‘ker α’ the linear subspace of P2 on which the map determined by α is not defined,‘im α’ will be the image of this map, and the rank ‘rk α’ of α will be 1+dim(im α).So:α ∈PGL(3) ⇐⇒ker α = ∅⇐⇒im α = P2 ⇐⇒rk α = 3.The curve C determines a rational mapc : P8 99K PNas follows: for α ∈P8, let c(α) be the curve defined by the degree-d polynomialequation F(α(x0, x1, x2)) = 0.So c(α) is defined as long as F(α(x0, x1, x2))doesn’t vanish identically; i.e., precisely if im α ̸⊂C.If α ∈PGL(3), then c(α) is the translate of C by α; therefore, c(PGL(3)) isjust the orbit OC of C in PN for the natural action of PGL(3).As an alternative description for the map c, consider for any point p ∈P2 theequationF(α(p)) = 0.As an equation ‘in p’, this defines the translate c(α); as an equation ‘in α’ thisdefines the hypersurface of P8 consisting of all α that map p to a point of C. Wewill call these hypersurfaces, that will play an important role in our discussion,‘point-conditions’. The rational map defined above is clearly the map defined bythe linear system generated by the point-conditions on P8.Our task here is to resolve the indeterminacies of the map c : P8 99K PN, by asequence of blow-ups at smooth centers: we will get a smooth projective varietyeV filling a commutative diagramPGL(3)⊂eVec−−−−→PNπyPGL(3)⊂P8c- - - -> PNThe image of eV in PN by ec will then be the orbit closure OC.

In §3 we will useec to pull-back questions about OC to eV ; the explicit description of eV obtained inthis section will enable us to answer these questions.The plan is to blow-up the support of the base locus of c; we will get a varietyV1 and a rational map c1 : V1 99K PN. We will then blow-up the support of the

4PAOLO ALUFFI CAREL FABERbase locus of c1, getting a variety V2 and a rational map c2 : V2 99K PN; in the casewe are considering here (i.e., the curve C is smooth to start with), repeating thisprocess yields eventually a variety eV as above. The support of the first base locusis in fact a copy of P2×C in P8 (see §2.1); if (k, q) ∈P2×C, and ci denotes the mapobtained at the i-th stage, we will find that ci still has indeterminacies over (k, q)if and only if the tangent line to C at q intersects C at q with multiplicity > i. So,for example, if C has only simple flexes then the map c3 is regular (Proposition2.9); and in general the number of blow-ups needed equals the highest possiblemultiplicity of intersection of a line with C.We should point out that (even for smooth C) this is not the only way toconstruct a variety eV as above: in fact, a different sequence of blow-ups is the onethat seems to generalize naturally to approach the same problem for singular C.§2.1.

The first blow-up. The set of rank-1 matrices in P8 is the image of theSegre embeddingˇP2 × P2 ֒→P8given in coordinates by((k0 : k1 : k2), (q0 : q1 : q2)) 7→k0q0k1q0k2q0k0q1k1q1k2q1k0q2k1q2k2q2where k0x0 + k1x1 + k2x2 = 0 is the kernel of the matrix, and (q0 : q1 : q2) is itsimage.

Intrinsically, this is just the map induced from the mapˇC3 ⊕C3 →ˇC3 ⊗C3 = Hom(C3, C3)(f, u) 7→f ⊗uWe have already observed that the map c : P8 99K PN is not defined at α ∈P8precisely when im α ⊂C; if C is smooth (therefore irreducible), this means thatthe image of α is a point of C. Therefore:the support of the base locus of c is the image of ˇP2 × C in P8 via the Segreembedding identifying ˇP2 × P2 with the set of rank-1 matrices.In particular, the support of the base locus of c is smooth, since C is. We letthen B = ˇP2 × C, and we let V1π1−→P8 be the blow-up of P8 along B. SinceB ∩PGL(3) = ∅, V1 contains a dense open set which we can identify with PGL(3).Also, the linear system generated by the proper transforms in V1 of the point-conditions (which we will call ‘point-conditions in V1’), defines a rational mapc1 : V1 99K PN making the diagramPGL(3)⊂V1c1- - - -> PNπ1yPGL(3)⊂P8c- - - -> PN

LINEAR ORBITS OF SMOOTH PLANE CURVES5commutative. The exceptional divisor E1 in V1 is the projectivized normal bundleof B in P8: E1 = P(NBP8).We will show now that the base locus of c1 issupported on a P1-subbundle of E1 over B.Let (k, q) be a point of B = ˇP2 × C: i.e., a rank-1 α ∈P8 with ker α = k,im α = q ∈C.

Also, let ℓbe the line tangent to C at q, let p be a point of P2, anddenote by P the point-condition in P8 corresponding to p.Lemma 2.1. (i) The tangent space to B at (k, q) consists of all ϕ ∈P8 such thatim ϕ ⊂ℓand ϕ(k) ⊂q.

(ii) P is non-singular at (k, q), and the tangent space to P at (k, q) consists ofall ϕ ∈P8 such that ϕ(p) ⊂ℓ.We are using our notations rather freely here. For example, in (i) α = (k, q) isin the tangent space since α(k) = ∅(as α is not defined along k).Proof.

(i) The tangent space to B at (k, q) is spanned by the plane {(k′, q) ∈B : k′ ∈ˇP2} = {ϕ ∈P8 : im ϕ = q} and by the line {(k, q′) ∈B : q′ ∈ℓ} ={ϕ ∈P8 : ker ϕ = k, im ϕ ∈ℓ}. Both these subspaces of P8 are contained in{ϕ ∈P8 : im ϕ ⊂ℓ, ϕ(k) ⊂q}; since this latter has clearly dimension 3, we aredone.

(ii) For α = (k, q) and ϕ ∈P8 consider the line α+ϕ t. Restricting the equationfor P to this line gives the polynomial equation in tF((α + ϕ t)(p)) = 0, i.e.F(α(p)) +Xi ∂F∂xiα(p)ϕi(p) t + · · · = 0(where ϕi(p) denotes the i-th coordinate of ϕ(p)).F(α(p)) = 0 since im α = q ∈C; the line is tangent to P at α when the linearterm also vanishes, i.e. if Pi(∂F/∂xi)qϕi(p) = 0.

This says precisely ϕ(p) ⊂ℓ, asclaimed.P is non-singular at α because any ϕ not satisfying the condition ϕ(p) ⊂ℓgivesa line α+ϕ t intersecting P with multiplicity 1 at α, by the above computation.□With the same notations, the tangent space to ˇP2×P2 at α consists of all ϕ withϕ(ker α) ⊂im α (intrinsically, all transformations ϕ inducing a map coim α −→coker α).The set of all ϕ such that im ϕ ⊂ℓforms (for any α) a 5-dimensional spacecontaining the tangent space to B at α, and therefore determines a 2-dimensionalsubspace of the fiber of NBP8 over α. As α moves in B we get a rank-2 subbundleof NBP8, and hence a P1-subbundle of E1 = P(NBP8), which we denote B1.Notice that B1 is non-singular, as a P1-bundle over the non-singular B.Proposition 2.2.

The base locus of the map c1 : V1 99K PN is supported on B1.Proof. Since c1 is defined by the linear system generated by all point-conditionsin V1, we simply need to show that the intersection of all point-conditions in V1 isset-theoretically B1.

This assertion can be checked fiberwise over α = (k, q) ∈B;

6PAOLO ALUFFI CAREL FABERso all we need to observe is that the intersection of the tangent spaces to all point-conditions at α consists (by Lemma 2.1 (ii)) of the ϕ ∈P8 such that ϕ(p) ⊂ℓforall p; i.e., the 5-dimensional space used above to define B1.□If P (p)1denotes the point-condition in V1 corresponding to p ∈P2, we have justshown Tp∈P2 P (p)1is supported on B1. The proof says a little more:Remark 2.3.

Tp∈P2 P (p)1∩E1 = B1 (scheme-theoretically).Indeed on each fiber of E1 (say over α ∈B) the fiber of B1, a linear subspace,is cut out by the fibers of the P (p)1∩E1, linear subspaces themselves; and thesituation clearly globalizes as α moves in B.§2.2.The second blow-up. Let V2π2−→V1 be the blow-up of V1 along B1.The new exceptional divisor is E2 = P(NB1V1); call ‘point-conditions in V2’ theproper transforms of the point-conditions of V1.

The linear system generated bythe point-conditions defines a rational map c2 : V2 99K PN; again, we obtain adiagramPGL(3)⊂V2c2- - - -> PNπ2◦π1yPGL(3)⊂P8c- - - -> PNand we proceed to determine the support of the base locus of c2.Let eE1 be the proper transform of E1 in V2. ThenLemma 2.4.

The base locus of c2 is disjoint from eE1.Proof. This is basically a reformulation of Remark 2.3: eE1 is the blow-up of E1along B1, and B1 is cut out scheme-theoretically by the intersections of E1 withthe point-conditions of V1.

So the intersection of the point-conditions in V2 mustbe empty along eE1, which is the claim.□Lemma 2.4 reduces the determination of the support of the base locus of c2 toa computation in P8. Denote by B the scheme-theoretic intersection of the point-conditions in P8, so the support of B is B.

For α ∈B, let thα(B) be the maximumlength of the intersection with B of the germ of a smooth curve centered at α andtransversal to B (the ‘thickness’ of B at α, in the terminology of [Aluffi2]).Lemma 2.5. The base locus of c2 is disjoint from (π2 ◦π1)−1α if thα(B) ≤2.Proof.

The base locus of c2 is the intersection of all point-conditions in V2, i.e. theset of all directions normal to B1 and tangent to all point-conditions in V1.

Letthen γ(t) be a smooth curve germ centered at a point of B1 above α, transversalto B1, and tangent to all point-conditions in V1. By Lemma 2.4, γ is transversalto E1; therefore π1(γ(t)) is a smooth curve germ centered at α and transversalto B.

Since γ(t) intersects all point-conditions in V1 with multiplicity 2 or more,π1(γ(t)) must intersect all point-conditions in P8 with multiplicity 3 or more; Bis the intersection of all point-conditions in P8, so this forces thα(B) ≥3.□Now the key computation is

LINEAR ORBITS OF SMOOTH PLANE CURVES7Lemma 2.6. If α = (k, q) ∈B, and ℓis the line tangent to C at q, then thα(B)equals the intersection multiplicity of ℓand C at q.Proof.

Let m be the intersection multiplicity of ℓand C at q. To show thα(B) ≥m,we just have to produce a curve normal to B and intersecting all point-conditionswith multiplicity at least m at α; such is the line α + ϕ t, with ϕ ∈P8 such thatim ϕ = ℓand ϕ(k) ̸= q.

Indeed, the last condition guarantees normality (Lemma2.1 (i)); and, for general p, q = α(p) and ϕ(p) span ℓ: so F((α + ϕ t)(p)) is justthe restriction of F to a parametrization of ℓ, and it must vanish exactly m timesat t = 0. Notice that these directions are precisely those defining B1.To show thα(B) ≤m, let γ(t) be any smooth curve germ normal to B andcentered at α; we have to show that γ intersects some point-condition with mul-tiplicity ≤m at α.

In an affine open of P8 containing α, writeγ(t) = α + ϕ t + . .

..The equation for the point-condition corresponding to p restricts on γ toF((α + ϕ t + . .

. )(p)) = F(α(p)) +Xi ∂F∂xiα(p)ϕi(p) t + · · · = 0,where ϕi(p) denotes the i-th coordinate of ϕ(p).The coefficient of tm in thisexpansion is(*)1m!Xi1,...,im∂mF∂xi1 · · · ∂ximα(p)ϕi1(p) · · · ϕim(p)+ terms involving derivatives of lower order,and to conclude the proof we have to show that for some p this term doesn’tvanish.To see this, observe that since ℓand C intersect with multiplicity exactly m atq, then the formXi1,...,im∂mF∂xi1 · · · ∂ximα(p)xi1 · · · ximdoesn’t vanish identically on ℓ; since ϕ(ker α) ̸⊂q (γ is normal to B), this impliesthat the summand1m!Xi1,...,im∂mF∂xi1 · · · ∂ximα(p)ϕi1(p) · · · ϕim(p)vanishes exactly d −m times along the line k = ker α (as a function of p).

Butsince all the other summands in (*) involve derivatives of order < m, they vanishwith order > d −m along k. Therefore the order of vanishing of (*) along k mustbe exactly d −m, and in particular (*) can’t be identically 0, as we claimed.□We adopt the following convention:

8PAOLO ALUFFI CAREL FABERDefinition. A point q of C is a ‘flex of order r’ if the line tangent to C at qintersects C at q with multiplicity r + 2.

We will say that q is a ‘flex’ of C ifr ≥1, and that q is a ‘simple flex’ if r = 1.Now we observe that there is a section s : B1 −→E2: for α1 ∈B1, let α =π1(α1) ∈B, say α = (k, q), and let ℓbe the line tangent to C at q.By theconstruction of B1, there is a matrix ϕ ∈P8 with im ϕ ⊂ℓsuch that α1 is theintersection of E1 and the proper transform of the line α+ϕ t in V1; now let s(α1)be the intersection of E2 and the proper transform of the line α + ϕ t in V2 (it isclear that s(α1) does not depend on the specific ϕ chosen to represent α1).Let B2 be the image via s of the set {α1 ∈B1 : q is a flex of C}. Thus B2consists of a number of smooth three-dimensional components, one for each flexof C: each component maps isomorphically to a P1-bundle over one of the planes{(k, q) ∈B : q is a flex of C}.Proposition 2.7.

The base locus of the map c2 : V2 99K PN is supported on B2.Proof. Let α1 ∈B1, and α = (k, q) the image of α1 in B, as above.

Consider theintersection of the base locus of c2 with the fiber π−12 (α1) ∼= P3. By Lemma 2.5and 2.6 this is empty if q is not a flex of C; even if q is a flex of C, the intersectionis a linear subspace of P3 missing a P2 (by Lemma 2.4), thus it consists of at mostone point.

Thus all we have to show is that s(α1) is in the base locus of c2 if qis a flex of C (of order r ≥1). But, as observed in the proof of Lemma 2.6, theline α + ϕ t determining α1 intersects each point-condition in P8 with multiplicityat least r + 2 ≥3; therefore the proper transform of α + ϕ t is tangent to allpoint-conditions in V1, and it follows that s(α1) ∈all point-conditions in V2, asneeded.□§2.3.The third blow-up.

Let V3π3−→V2 be the blow-up of V2 along B2.The new exceptional divisor is E3; the ‘point-conditions of V3’ are the propertransforms of the point-conditions of V2.The linear system generated by thepoint-conditions defines a rational map c3 : V3 99K PN, making the diagramPGL(3)⊂V3c3- - - -> PNπ3◦π2◦π1yPGL(3)⊂P8c- - - -> PNcommute. We will show now that c3 is a regular map if all the flexes of C aresimple, so that in this case V3 is the variety we are looking for.

For each flex oforder > 1, we will find a four-dimensional component in the base locus of c3, andmore blow-ups will be needed.Call B2 the scheme-theoretic intersection of the point-conditions in V2, so B2is supported on B2. For α2 ∈B2, define the thickness thα2(B2) of B2 at α2 as wedid above for thα(B).

Also, let α = (k, q) be the image of α2 in B. With thesenotations:

LINEAR ORBITS OF SMOOTH PLANE CURVES9Lemma 2.8. If q is an flex of order r of C, then thα2(B2) = r.Proof.

We have to show that if γ(t) is a smooth curve germ in V2, centered at α2and transversal to B2, then the maximum length of the intersection of B2 and γat t = 0 is precisely r.Suppose first that γ is transversal to E2: then, as argued in the proof of Lemma2.5, the image of γ in P8 is a smooth curve germ centered at α and transversal toB: by Lemma 2.6, the length of the intersection of B and this curve is at mostr + 2; it follows that the maximum length of the intersection of B2 and such γ’sis indeed r (attained for example by the proper transform of α + ϕ t, with ϕ as inthe proof of Lemma 2.6).Thus we may assume that γ is tangent to E2, and we have to show thatClaim. B2 ∩γ(t) vanishes at most r times at t = 0.This is a lengthy but straightforward coordinate computation, which we leaveto the reader.

The outcome is that the maximum length is r, and it is attainedin the direction normal to B2 in the section s(B1) ⊂E2 defined in §2.2.□The next results are now easy consequences.Proposition 2.9. If all flexes of C are simple, then the map c3 : V3 99K PN isregular.Proof.

We have to show that c3 has no base locus, i.e. that the intersection ofall point-conditions in V3 is empty.

But a point in the intersection of all point-conditions in V3 would determine a direction normal to B2 and tangent to allpoint-conditions in V2; the thickness of B2 would then be ≥2 at some point. ByLemma 2.8, if all flexes of C are simple (i.e., of order 1) the thickness of B2 isprecisely 1 everywhere on B2, so this can’t happen.□By Proposition 2.9, we are done in the case when C has only simple flexes: V3is the variety eV we meant to construct.

We will show now that for each flex ofC of order r > 1, the base locus of c3 has a smooth four-dimensional connectedcomponent.Let α2 ∈B2, mapping to α = (k, q) in B, and assume q is a flex of C of orderr > 1. B2 is 3-dimensional, so the fiber π−13 (α2) of E3 = P(NB2V2) over α2 is a P4.We have two special points in this P4, namely the point determined by the propertransform of the line α + ϕ t used in §2.2 to define s, and the direction normal toB2 in the section s(B1).

We have seen in the proof of Lemma 2.8 that the lengthof the intersection of these directions with B2 is exactly r; also, these points aredistinct for all α2 (since one of them corresponds to a direction contained in E2,while the other comes from a direction transversal to E2), so they determine a P1in the fiber π−13 (α2). As α2 moves in the component of B2 over q, this P1 traces aP1-bundle over that component, a smooth four-dimensional subvariety B(q)3of E3.Call B3 the union of all these (disjoint) subvarieties of E3, arising from non-simpleflexes of C.Proposition 2.10.

The base locus of the map c3 : V3 99K PN is supported on B3.Proof. The argument here is somewhat analogous to the argument in the proof of2.7.

We have to show that in each fiber π−13 (α2) ∼= P4 as above, the intersection

10PAOLO ALUFFI CAREL FABERof all point-conditions is supported on the specified P1. Observe that each point-condition determines a hyperplane in this P4, so that the intersection of the baselocus of c3 with π−13 (α2) must be a linear subspace of this P4.Secondly, forthe same reason, no directions tangent to the fiber of E2 containing α2 can betangent to all point-conditions in V2.

The fibers of E2 are three-dimensional andtransversal to B2, thus this shows that the base locus of c3 must miss a P2 in thefiber π−13 (α2). Thus, the intersection of the base locus of c3 with π−13 (α2) canconsist of at most a P1.Therefore, we just have to show that the two points of π−13 (α2) used in theconstruction of B3 are contained in all point-conditions of V3; or, equivalently, thetwo directions in V2 used to define these points are tangent to all point-conditionsin V2.

But this is precisely the result of the computation in the proof of Lemma2.8: the length of the intersection of these curves with all point-conditions isr ≥2.□§2.4 Further blow-ups. As we have seen in §2.3, each non-simple flex q of Cgives rise to a smooth four-dimensional component of the support B3 of the baselocus of c3; and B3 is the union of all such components.The plan is still toblow-up the support of the base-locus; since the components are disjoint, we canconcentrate on a specific one: say B(q)3 , corresponding to a flex q of C of orderr ≥2.Let V (q)3be the complement of all components of B3 other than B(q)3in V3.

LetV (q)4−→V (q)3be the blow-up of V (q)3along B(q)3 ; again, the proper transforms inV (q)4of the point-conditions define a map c(q)4: V (q)499K PN. The base locus ofc(q)4might have components over B(q)3 , whose union we denote B(q)4 ; in this case,we will let V (q)5be the blow-up of V (q)4along B(q)4 .

Iterating this process we get atower of varieties and maps:.........B(q)i+1⊂V (q)i+1c(q)i+1- - - -> PNyyB(q)i⊂V (q)ic(q)i- - - -> PNyy.........B(q)3⊂V (q)3c3- - - -> PNwhere, inductively for i ≥4: V (q)i−→V (q)i−1 is the blow-up of V (q)i−1 along B(q)i−1;c(q)i: V (q)i99K PN is defined by the proper transforms in V (q)iof the point-conditions (i.e., the ‘point-conditions in V (q)i’); and (for i ≥3) B(q)iis the support

LINEAR ORBITS OF SMOOTH PLANE CURVES11of the intersection B(q)iof the point-conditions in V (q)i(i.e., the base locus of c(q)i ).Also, for i ≥3 let E(q)ibe the exceptional divisor in V (q)i, and let eE(q)ibe theproper transform of E(q)iin V (q)i+1.Lemma 2.11. If q is a flex of order r ≥2, then for 3 ≤i ≤r + 1:(1)i: V (q)iis non-singular(2)i: the composition map B(q)i−→B(q)3is an isomorphism(3)i: the thickness of B(q)iis r + 2 −i at each point of B(q)i(4)i: B(q)i+1 ∩eE(q)i= ∅Proof.

We have (1)3, (2)3 trivially, and (3)2 by Lemma 2.8. Also, since B3 is cutout by linear spaces in each fiber of E3, we have (4)3.

Now we will show that:Claim. For 4 ≤i ≤r + 1, (1)i−1, (2)i−1, (3)i−2 and (4)i−1 imply (1)i, (2)i,(3)i−1, and (4)i.Also, we will show that (3)r, (4)r+1 imply (3)r+1: this will prove the statement.Proof of the Claim.

In this proof we will drop the (q) notation, to ease the expo-sition. Vi is then the blow-up of Vi−1 along Bi−1, and these are both non-singularby (1)i−1, (2)i−1: so Vi must also be non-singular, giving (1)i.Next, compute the thickness of Bi−1:let γ(t) be any smooth curve germtransversal to Bi−1 and centered at any αi−1 ∈Bi−1.

If γ is tangent to Ei−1,then by (4)i−1 its proper transform will miss the general point-condition in Vi:i.e., the length of the intersection of γ(t) with Bi−1 at t = 0 is 1. If γ is transversalto Ei−1 (and Bi−1), then γ maps down to a smooth curve germ γ∗centered at apoint of Bi−2 and transversal to Bi−2.

By (3)i−2, the intersection of γ∗with thepoint-conditions in Vi−2 has length at most r −i + 4: it follows that the intersec-tion of γ with the point-conditions in Vi−1 has length at most r −i + 3 ≥2 (sincei ≤r + 1). Therefore the thickness of Bi−1 at αi−1 is r −i + 3, which gives (3)i−1.For (2)i, look at the intersection of Bi with the fiber of Ei over an arbitraryαi−1 ∈Bi−1.

First we argue this can’t be empty: indeed, thαi−1(Bi−1) = r−i+3 ≥2, so through every αi−1 in Bi−1 there are directions tangent to all point-conditionsin Vi−1. To get (2)i, we need to show that the fiber of Bi over αi−1 consists(scheme-theoretically) of a simple point.

But this is the intersection of Bi withthe fiber of Ei (∼= P3) over αi−1, thus a nonempty intersection of linear subspacesin P3 missing a hyperplane (by (4)i−1): precisely a point, as needed for (2)i.Finally, we need (4)i. Once more observe that Bi intersects each fiber of Ei inan intersection of linear spaces: thus there are no directions in the fibers of Ei andtangent to all point-conditions in Vi.

This says that Bi+1 must avoid the propertransforms in Vi+1 of all fibers of Ei, and therefore eEi, giving (4)i.This proves the Claim.The only case not covered yet is (3)r+1: to obtainthis and conclude the proof of 2.11, apply the same argument as above to (3)r,(4)r+1.□Lemma 2.11 describes the sequence of blow-ups over V3 that takes care of aspecific flex q on C of order r ≥2. The case i = r+1 of the statement says that thevariety V (q)r+1 is non-singular, and the base locus of the map c(q)r+1 : V (q)r+1 99K PN is

12PAOLO ALUFFI CAREL FABERsupported on a variety B(q)r+1 isomorphic to B(q)3 ; moreover, for all αr+1 ∈B(q)r+1, wegot thαr+1(Br+1) = 1. Let then V (q)r+2 −→V (q)r+1 be the blow-up of V (q)r+1 along B(q)r+1,and denote by c(q)r+2 the rational map V (q)r+2 99K PN defined by the point-conditionsin V (q)r+2.

Then V (q)r+2 is clearly non-singular, andCorollary 2.12. c(q)r+2 is a regular map.Proof. Indeed, the point-conditions in V (q)r+2 cannot intersect anywhere along E(q)r+2:if they did, any intersection point would correspond to a direction normal to B(q)r+1and tangent to all point-conditions in V (q)r+1, and the thickness of B(q)r+1 would be≥2, in contradiction with Lemma 2.11.□By this last result, the sequence of r−1 blow-ups over V3 just described resolvesthe indeterminacies of c3 : V3 99K PN over the component B(q)3of B3.

To resolveall indeterminacies of c3, we just have to apply the construction simultaneouslyto all components of B3: build the sequence.........Bi+1⊂Vi+1ci+1- - - -> PNyyBi⊂Vici- - - -> PNyy.........B3⊂V3c3- - - -> PNwhere, for i ≥4, Vi −→Vi−1 is the blow-up of Vi−1 along Bi−1, ci : Vi 99K PNis defined by the proper transforms in Vi of the point-conditions, and Bi is thesupport of the base locus of ci. By Lemma 2.11 and Corollary 2.12 all Vi’s are non-singular, and, for each flex q of C of order r, Bi has either exactly one componentmapping isomorphically to B(q)3if i ≤r+1, or no component over B(q)3if i ≥r+2.In particular, this construction will stop!

If r is the maximum among the orderof the flexes of C, let eV = Vr+2, ec = cr+2, and let π be the composition of ther + 2 blow-up maps; then we have shownTheorem II. ec : eV −→PN is a regular map, and the diagramPGL(3)⊂eVec−−−−→PNπyPGL(3)⊂P8c- - - -> PNcommutes.which was our objective.

LINEAR ORBITS OF SMOOTH PLANE CURVES13§3. The degree of the orbit closureIn this section we employ the blow-up construction of §2 to compute the degreeof the orbit closure OC of a smooth plane curve C ∈PN = Pd(d+3)2with at mostfinitely many automorphisms (if d = 3, we should specify ‘induced from PGL(3)’.This will be understood in the following).

The degree will depend on just sixnatural numbers: the order of the group of automorphisms of C, the degree d ofC, and four numbers encoding information about the number and order of theflexes of C.In fact, the blow-up construction of §2 yields most naturally the‘predegree’ of OC:Definition. The ‘predegree’ of OC is the 8-fold self-intersection eP 8 of the classeP of a point-condition in eV .Lemma 3.1.

The predegree of OC equals the product of the degree of the orbitclosure of C by the order oC of the group of automorphisms of C induced fromPGL(3).Proof. The map ec is defined by the linear system generated by the point-conditionson eV , so eP is the pull-back of the hyperplane class from PN.Therefore eP 8computes the pull-back of the intersection of ec(eV ) = OC with 8 hyperplanes ofPN: i.e., the product of deg(OC) by the degree of the map ec.

This latter equalsoC since, given a general c(α) ∈OC (α ∈P8), the fiber of c(α) consists of allproducts ϕα, where ϕ fixes C.□Observe that for the general C of degree ≥4, the predegree of OC equals thedegree of the orbit closure. Our aim here is to compute the predegree of OC,by using the construction of eV described in §2: we will show that this numberdepends only on d and on the type of the flexes of C.Our tool will be a formula relating intersection degrees under blow-ups:Proposition 3.2.

Let Bi֒→V be non-singular projective varieties, and let X ⊂Vbe a codimension-1 subvariety, smooth along B. Let eV be the blow-up of V alongB, and let eX be the proper transform of X. ThenZeV[ eX]dim V =ZV[X]dim V −ZB([B] + i∗[X])dim Vc(NBV )whereReV , etc.

denote the degree of a class in eV , etc., cf. [Fulton], Def.

1.4. Note:we will omit the R sign and the class brackets [·] when this doesn’t create ambi-guities.Proof.

This follows from [Aluffi1], §2, Theorem II and Lemma (2), (3).□We will compute the predegree of OC (i.e. eP 8) by applying Proposition 3.2 toeach blow-up in the sequence giving eV : the missing ingredients to be obtainedat this point are the Chern classes of the normal bundles of the centers of theblow-ups, and calculations in their intersection rings.

14PAOLO ALUFFI CAREL FABERIn the following, P, Pi, eP will denote resp. (the class of) point-conditions inV, Vi, eV .

The embedding of Bj in Vj is denoted ij, and pjk will be used for themap Bj −→Bk (pj will be pjj−1 for short). As a general convention, we will omitpull-back notations unless we fear ambiguity.§3.1.

The first blow-up. The center of the first blow-up is the variety B =ˇP2 × C; the embedding i : B ֒→P8 is given by composition with the Segreembedding:B = ˇP2 × C ⊂ˇP2 × P2 −→P8.Call h, k resp.

the hyperplane class in P2, ˇP2. Our convention on pull-backs allowsus to write k, h for the pull-backs of k, h from the factors to ˇP2 × P2, and toB ⊂ˇP2 × P2.Also, since the Segre embedding is linear on each factor, thehyperplane class of P8 pulls-back to k + h on B.Lemma 3.3.

If C has degree d:(i) In B: k3 = 0, k2h = d, kh2 = 0, h3 = 0(ii) c(NBP8) = (1 + k + h)9(1 + dh)(1 + k)3(1 + h)3(iii) P 8 = d8; and P pulls-back to dk + dh.Proof. (i) is immediate.

(ii) c(NBP8) = c(NBˇP2 × P2)c(NˇP2×P2P8) by the Whitney formula and theexact sequence of normal bundles. Now, since B = ˇP2 × C, c(NBˇP2 × P2) =c(NCP2) = 1 + dh.

The formula for c(NˇP2×P2P8) is standard. (iii) Recall from §2 that if p ∈P2, P is the point-condition corresponding top, and F(x0 : x1 : x2) is the (degree-d) polynomial defining C, then α ∈P ⇐⇒F(α(p)) = 0: so P is defined by a degree-d equation in P8.□We have already observed that the point-conditions are non-singular (Lemma2.1 (ii)), so we are ready for the key computation needed to apply Proposition 3.2to the first blow-up:Lemma 3.4.ZB(B + i∗P)8c(NBP8)= d(10d −9)(14d2 −33d + 21)Proof.

By Lemma 3.3, this isZˇP2×C(1 + dk + dh)8(1 + k)3(1 + h)3(1 + k + h)9(1 + dh):the statement follows by computing the coefficient of k2h (the only term withnon-zero degree, by Lemma 3.3(i)).□

LINEAR ORBITS OF SMOOTH PLANE CURVES15§3.2. The second blow-up.

The center of the second blow-up is a P1-bundleB1 over BB1i1−−−−→V1p1yyBi−−−−→P8so classes on B1 are combinations of (the pull-backs of) k, h and c1(OB1(−1));we call this latter e, and observe it is the pull-back from V1 of the class of theexceptional divisor E1.Lemma 3.5. (i) p1∗ei =0−1−3k + 2dh −6h−6k2 + 9dkh −27kh24dk2h −72k2hi = 0i = 1i = 2i = 3i = 4(ii) c(NB1V1) = (1 + e)(1 + k + dh −e)3(iii) i∗1P1 = dk + dh −eProof.

(iii) is immediate, as P is non-singular and pulls-back on B to dk + dh(Lemma 3.3 (iii)).For (i) and (ii) we need to produce B1 ⊂E1 more explicitly as the projectiviza-tion of a rank-2 subbundle of NBP8.First define for any p ∈P2 a rank-8 subbundle Hp of the trivial bundle B × C9over B: if F is a polynomial defining C, and (k, q) ∈B, A ∈C9 = Hom(C3, C3),say((k, q), A) ∈Hp ⇐⇒2Xi=0 ∂F∂xiqA(p)i = 0where A(p)i is the i-th coordinate of A(p).So the fiber of Hp over q is thehyperplane of matrices A ∈C9 such that A(p) ∈line tangent to C at q. Noticethat the above equation has degree d −1 in the coordinates of q: thus (denotingby C9 the trivial bundle B × C9, for short)c1 C9Hp= (d −1)h.Now restrict the Euler sequence for P8 to B via Bi֒→P8: Hp ⊂C9 determinesa subbundle Hp of i∗T P8 and we have the following diagram of bundles over B

16PAOLO ALUFFI CAREL FABER(suppressing pull-back as usual)000yyy0 −−−−→O −−−−→Hp ⊗OP8(1) −−−−→Hp−−−−→0yy0 −−−−→O −−−−→C9 ⊗OP8(1) −−−−→T P8 −−−−→0yyy0 −−−−→C9Hp⊗OP8(1)T P8Hp−−−−→0yy00from which it followscT P8Hp= c C9Hp⊗OP8(1)= 1 + k + dh.Also, observe that each Hp contains T B.Now let p1, p2, p3 be non-collinear points. A matrix has image contained ina line if and only if it sends three non-collinear points to that line, thus theintersection Hp1 ∩Hp2 ∩Hp3 is the rank-6 bundle over B = ˇP2 × C whose fiberover (k, q) ∈B consists of all matrices whose image is contained in the line tangentto C at q.

This is the space we used to define B1: if we set Q = Hp1 ∩Hp2 ∩Hp3,thenB1 = P QT B⊂P(NBP8) = E1,andcT P8Q= (1 + k + dh)3.Finally, the Euler sequences for E1 and B1 give the diagram000yyy0 −−−−→O −−−−→QT B ⊗OB1(1)−−−−→T B1|B −−−−→0yy0 −−−−→O −−−−→NBP8 ⊗OB1(1) −−−−→T E1|B −−−−→0yyy0 −−−−→T P8Q⊗OB1(1)NB1E1 −−−−→0yy00

LINEAR ORBITS OF SMOOTH PLANE CURVES17(here T B1|B, T E1|B denote the relative tangent bundles of B1, E1 over B) fromwhichc(NB1E1) = cT P8Q⊗OB1(1)= (1 + k + dh −e)3.From this discussion, it’s easy to obtain (i) and (ii):(i) p1∗Xi(−1)iei = c QT B−1by [Fulton], Proposition 3.1 (a)= cT P8Qc(NBP8)−1by Whitney’s formula= (1 + k + dh)3(1 + k)3(1 + h)3(1 + k + h)9(1 + dh)by the above and Lemma 3.3 (ii)= 1 −3k + 2dh −6h + 6k2 −9dkh + 27kh + 24dk2h −72k2h. (ii) c(NB1V1) = c(NE1V1)c(NB1E1) = (1 + e)(1 + k + dh −e)3.□Lemma 3.5 allows us to compute the term needed to apply Proposition 3.2 tothe second blow-up:Lemma 3.6.ZB1(B1 + i∗1P1)8c(NB1V1)= d(2d −3)(322d2 −1257d + 1233)Proof.

This isZB1(1 + dk + dh −e)8(1 + e)(1 + k + dh −e)3by Lemma 3.5 (ii) and (iii). Since the degree doesn’t change after push-forwards,this is alsoZBp1∗(1 + dk + dh −e)8(1 + e)(1 + k + dh −e)3.Computing the degree-4 term in the expansion of the fraction and applying Lemma3.5 (i) and the projection formula, this is computed as a sum of degree-3 terms ink, h over B. Lemma 3.3 (i) is used then to obtain the stated expression.□§3.3.The third blow-up.

At this point we have to start taking flexes intoaccount. For any q ∈C, let fℓ(q) be the order of q as a flex of C, in the sense of§2.2: so fℓ(q) = 0 if q is not a flex of C, fℓ(q) = 1 if q is a simple flex of C, andso on.The center B2i2֒→V2 of the third blow-up is the disjoint unionB2 =[fℓ(q)>0B(q)2,

18PAOLO ALUFFI CAREL FABERwhere each B(q)2maps isomorphically to the restriction B(q)1of the P1-bundle B1to ˇP2 × {q} ⊂B. Moreover, B2 ∩eE1 = ∅(Lemma 2.4).

As h restricts to 0 oneach ˇP2 × {q}, the intersection ring of B(q)2is generated by k, e (defined as in§3.2). Also, we denote by e′ the pull-back of E2 to B(q)2 , and by p20 the mapB(q)2−→ˇP2 × {q} ∼= P2.Lemma 3.7.

(i) e′ = e(ii) p20∗ei =0−1−3k−6k2i = 0i = 1i = 2i = 3(iii) c(NB(q)2 V2) = (1 + e)(1 + k −2e)3(iv) i∗2P2 = dk −2eProof. (ii) follows from Lemma 3.5 (i), since the restriction of h to B(q)2is 0.The key observation for the other points is that B(q)2∩eE1 = ∅.RealizeB(q)2⊂P(NB1V1) as P(L), where L is a sub-line bundle of NB1V1.eE1 ∩E2 isthe exceptional divisor of the blow-up of E1 along B1, i.e.

the projectivization ofNB1E1 in NB1V1. That P(L) and P(NB1E1) are disjoint says that L ∩NB1E1 isthe zero-section of NB1V1, and thereforeL ∼= NB1V1NB1E1= NE1V1as bundles on B(q)1 .

(i) With the same notations, L is tautologically the universal line bundle overP(L); it must then equal the restriction to B(q)2of the universal line bundleOE2(−1) ∼= NE2V2. In other wordsL ∼= NE2V2as bundles on B(q)2 .Since the projection from B(q)2to B(q)1is an isomorphism, it follows thate = c1(NE1V1) = c1(L) = c1(NE2V2) = e′.

(iii) Call E(q)2the restriction of E2 = P(NB1V1) to B(q)1 . We have Euler se-quences0 −−−−→O −−−−→L ⊗O(1)−−−−→T B(q)2 |B(q)1−−−−→0yyy0 −−−−→O −−−−→NB1V1 ⊗O(1) −−−−→T E(q)2 |B(q)1−−−−→0and we just argued L ∼= O(−1): soc(NB(q)2 E(q)2 ) = cNB1V1L⊗ˇL(restricted to B(q)2 )= (1 + e −e′)(1 + k −e −e′)3(1 + e′ −e′)= (1 + k −2e)3by (i);

LINEAR ORBITS OF SMOOTH PLANE CURVES19next, since NB(q)1 B1 is clearly trivial, we have c(NE(q)2 E2) = 1; so putting NB(q)2 V2together:c(NB(q)2 V2) = c(NE2V2)c(NE(q)2 E2)c(NB(q)2 E(q)2 ) = (1 + e)(1 + k −2e)3,as claimed. (iv) Since P1 is non-singular along B1, P2 restricts to dk −e −e′ = dk −2e by(i).□We are ready for the term needed to apply Proposition 3.2 to the third blow-up:Lemma 3.8.ZB2(B2 + i∗2P2)8c(NB2V2)=Xfℓ(q)>0(196d2 −960d + 1125)Proof.

By Lemma 3.7 (iii) and (iv), this isXfℓ(q)>0ZB(q)2(1 + dk −2e)8(1 + e)(1 + k −2e)3 =Xfℓ(q)>0ZP2 p20∗(1 + dk −2e)8(1 + e)(1 + k −2e)3(pushing forward doesn’t change degrees) and one concludes with the projectionformula and Lemma 3.7 (ii).□§3.4. Further blow-ups.

Further blow-ups are necessary if there are points qon C with fℓ(q) > 1. We first attack the initial step.The center B3i3֒→V3 of the fourth blow-up is the disjoint unionB3 =[fℓ(q)>1B(q)3,where each B(q)3is a P1-bundle over B(q)2 .

The intersection ring of B(q)3is generatedby (the pull-back of) the classes k, e of B(q)2 , and by the class of the universalline bundle, i.e. the pull back f of E3 from V3.Denote by p3 the projectionB(q)3−→B(q)2 .Lemma 3.9.

(i) p3∗f i =0−1−e−e2−e3i = 0i = 1i = 2i = 3i = 4(ii) c(NB(q)3 V3) = (1 + f)(1 + k −2e −f)3(iii) i∗3P3 = dk −2e −fProof. (iii) is clear, as P2 is non-singular along B(q)3 .

20PAOLO ALUFFI CAREL FABERFor the other items, we have to produce B(q)3⊂E(q)3= P(NB(q)2 V2) explicitlyas the projectivization of a rank-2 subbundle of NB(q)2 V2. Recall that each fiberof B(q)3is spanned by two points corresponding respectively to (1) a directiontransversal to E2, and (2) a direction in E2, transversal to the fiber of E2.

Sincethese two points are always distinct, B(q)3= P(L1 ⊕L2), where PL1, PL2 give thetwo distinguished points on each fiber. Now, L1 ∩NB(q)2 E2 is the zero-section inNB(q)2 V2 (the first direction is transversal to E2); so, with L as in the proof of 3.7,L1 ∼= NE2V2 ∼= L.Similarly, since the second direction is transversal to the fiber of E2, whose normalbundle in E2 is trivial, L2 ∼= O; and therefore we haveB(q)3= P(L ⊕O).

(i) As in the proof of 3.5 (i),p3∗Xi(−1)if i = c(L ⊕O)−1 =Xi(−1)ieiand (i) follows by matching dimensions. (ii) Another pair of Euler sequences: on B(q)30 −−−−→O −−−−→(L ⊕O) ⊗O(1) −−−−→T B(q)3 |B(q)2−−−−→0yyy0 −−−−→O −−−−→NB(q)2 V2 ⊗O(1) −−−−→T E(q)3 |B(q)2−−−−→0Since c1(O(1)) = −f and E3 is the disjoint union of the E(q)3 :c(NB(q)3 E3) = c(NB(q)3 E(q)3 )= c NB(q)2 V2L ⊕O ⊗O(1)!= (1 + k −2e −f)3(the Chern roots of NB(q)2 V2 are e, k −2e, k −2e, k −2e, 0 by Lemma 3.7 (iii)).Finally:c(NB(q)3 V3) = c(NE3V3)c(NB(q)3 E3) = (1 + f)(1 + k −2e −f)3as stated.□

LINEAR ORBITS OF SMOOTH PLANE CURVES21Lemma 3.9 describes the situation at the fourth blow-up. The next blow-upsare built on this in the sequence described in §2.4: the center Bjij֒→Vj of the(j + 1)-st blow-up (j ≥3) is the disjoint unionBj =[fℓ(q)>j−2B(q)j,where each B(q)jmaps isomorphically down to B(q)3 , and is disjoint from eEi−1(Lemma 2.11).

The intersection ring of each B(q)j∼= B(q)3is then generated byk, e, f, and the relations stated in Lemma 3.9 (i) hold, for the projection pj2 :B(q)j−→B(q)2 .Denote by fj the pull-back of Ej to B(q)j ; Lemma 3.9 can beextended to all stages in the sequence:Lemma 3.9 (continued). For 3 ≤j ≤fℓ(q) + 1(i)j fj = f(ii)j c(NB(q)j Vj) = (1 + f)(1 + k −2e −(j −2)f)3(iii)j i∗jPj = dk −2e −(j −2)fProof.

For j = 3 this is given by Lemma 3.9. So it suffices to show that, for 3 ≤j ≤fℓ(q), (i)j, (ii)j, (iii)j imply (i)j+1, (ii)j+1, (iii)j+1.

Consider then B(q)j+1 =P(Lj+1) ⊂P(NB(q)j Vj). So fj+1 is the class of OB(q)j+1(−1), i.e.

of Lj+1. SinceB(q)j+1 ∩eEj = ∅(Lemma 2.11 (iv)), we get by the usual argumentfj+1 = c1(Lj+1) = c1(NEjVj) = fj:and fj = f by (i)j; so fj+1 = f, giving (i)j+1.

(iii)j+1 follows then from (iii)j and (i)j+1, since Pj is non-singular along Bj.Finally, we use the Euler sequences0 −−−−→O −−−−→Lj+1 ⊗O(1)−−−−→T B(q)j+1|B(q)j−−−−→0yyy0 −−−−→O −−−−→NB(q)j Vj ⊗O(1) −−−−→T E(q)j+1|B(q)j−−−−→0to get (since Ej+1 is the disjoint union of the E(q)j+1)c(NB(q)j+1Ej+1) = c(NB(q)j+1E(q)j+1)= c NB(q)j VjLj+1⊗O(1)!= (1 + f −f)(1 + k −2e −(j −2)f −f)3(1 + f −f)by (ii)j= (1 + k −2e −(j −1)f)3,

22PAOLO ALUFFI CAREL FABERsoc(NB(q)j+1Vj+1) = c(NEj+1Vj+1)c(NB(q)j+1Ej+1) = (1 + f)(1 + k −2e −(j −1)f)3,i.e. (ii)j+1.□We get then the key term to apply Proposition 3.2 to the j-th blow up in thesequence.

In fact, we can cover Lemma 3.8 as well in one statement:Lemma 3.10. For j ≥2ZBj(Bj + i∗jPj)8c(NBjVj)=Xfℓ(q)>j−230j4 −96(d −1)j3+ 12(d −1)(7d −11)j2 + 84(d −1)2j −7(2d −3)(22d −39).Proof.

For j = 2, this is Lemma 3.8. For j ≥3, by Lemma 3.9 this isXfℓ(q)>j−2ZB(q)j(1 + dk −2e −(j −2)f)8(1 + f)(1 + k −2e −(j −2)f)3.If pj2 denotes the projection B(q)j−→B(q)2 , (and p20 is the map B(q)2−→ˇP2 ×{q} ∼=P2, as in §3.3), this can be computed asXfℓ(q)>j−2ZP2 p20∗pj2∗(1 + dk −2e −(j −2)f)8(1 + f)(1 + k −2e −(j −2)f)3,which is evaluated by using the projection formula, 3.9 (i) and 3.7 (ii).□§3.5.

The predegree of OC. Computing the predegree of OC is now a straight-forward application of Proposition 3.2 and Lemmas 3.4, 3.6 and 3.10: by Propo-sition 3.2eP 8 = P 8 −Xj≥0ZBj(Bj −i∗jPj)8c(NBjVj)(where B0 = B, etc.

), and the terms in the summation have been computed insections 3.1–3.4. This givesProposition 3.11.

The predegree of OC isd8 −d(10d −9)(14d2 −33d + 21) −d(2d −3)(322d2 −1257d + 1233)−Xj≥2Xq∈Cfℓ(q)>j−2(30j4 −96(d −1)j3 + 12(d −1)(7d −11)j2+ 84(d −1)2j −7(2d −3)(22d −39)).This result can be given in handier forms. For example:

LINEAR ORBITS OF SMOOTH PLANE CURVES23Theorem III(a). The predegree of OC isd(d −2)(d6 + 2d5 + 4d4 + 8d3 −1356d2 + 5280d −5319) −Xq∈Cfℓ(q)(fℓ(q) −1)6fℓ(q)3 + (75 −24d)fℓ(q)2 + (28d2 −240d + 393)fℓ(q) + 196d2 −960d + 1125Proof.

Invert the order of the summations in Proposition 3.11, then use the factthat Pq∈C fℓ(q) = 3d(d −2) (the number of flexes of C, counted with multiplic-ity).□Or, in another form:Theorem III(b). Denote by f (r)Cthe sum Pq∈C fℓ(q)r. Then the predegree ofOC isd8 −8d(98d3 −492d2 + 843d −486) −(168d2 −720d + 732)f (2)C−(28d2 −216d + 318)f (3)C−(69 −24d)f (4)C−6f (5)C .By Theorem III(B), if C is smooth then the predegree of OC depends only onthe degree d of C and on the four numbers f (2)C , f (3)C , f (4)Cand f (5)C .If C only has simple flexes, then fℓ(q) = 0 or 1 for all q ∈C, so Theorem III(A)givesCorollary.

If all flexes of C are simple, then the predegree of OC isd(d −2)(d6 + 2d5 + 4d4 + 8d3 −1356d2 + 5280d −5319)= d8 −1372d4 + 7992d3 −15879d2 + 10638d.Denoting this polynomial in d by P(d), we remark that it gives the degree ofthe orbit closure of the general smooth plane curve of degree d ≥4 (indeed, sucha curve C has no non-trivial automorphisms, so by Lemma 3.1 the degree of OCequals the predegree).Remark. Denoting by fk(d) the (negative) contribution to the predegree aris-ing from a flex of order k on a curve of degree d, we have, as an immediateconsequence of Theorem III(A):fk(d) = −k(k−1)((28k+196)d2−(24k2+240k+960)d+(6k3+75k2+393k+1125)).E.g., f2(d) = −6(84d2 −512d + 753) and f3(d) = −6(280d2 −1896d + 3141).

Itis an easy calculus exercise to show that fk(d) < 0 whenever d ≥k + 2 ≥4. Thisproves that the predegree is maximal for a curve with only simple flexes.

24PAOLO ALUFFI CAREL FABER§3.6. Examples.

It is a consequence of Lemma 3.1 that the predegree of theorbit of a smooth plane curve is divisible by the order of its PGL(3)-stabilizer.This cuts both ways. On the one hand, each curve with non-trivial automorphismsprovides us with a non-trivial check of the formulas above.

On the other hand,these formulas might help in determining which automorphism groups of smoothplane curves occur. We illustrate this below.Consider, for d ≥3, the Fermat curve xd+yd+zd.

Its 3d flexes have order d−2,so the predegree of its orbit is P(d) + 3d · fd−2(d). So for each d this number isdivisible by 6d2, the order of the stabilizer.

This implies that in the ring Z[d] thepolynomial P(d)+ 3d·fd−2(d) is divisible by d2 and that the quotient polynomialtakes values divisible by 6. IndeedP(d) + 3d · fd−2(d) = d2(d −2)(d5 + 2d4 −26d3 −7d2 + 192d −192).Dividing this by 6d2, we get the degree of the orbit closure of the Fermat curve,i.e., of the trisecant variety to the d-uple embedding of P2 in PN, as mentioned inthe introduction.Here is a similar example for all d ≥5: the curve xd−1y + yd−1z + zd−1x.

Thepoints (1 : 0 : 0), (0 : 1 : 0) and (0 : 0 : 1) are flexes of order d −3; counted withmultiplicity, 3(d2 −3d + 3) flexes remain. The group D of diagonal matrices withentries (1, ζ, ζ2−d), where ζ is a (d2 −3d + 3)-rd root of unity, acts on the latterflexes without fixed points; so either there is one orbit of flexes of order 3, or oneorbit of flexes of order 2 and one orbit of simple flexes, or, finally, three orbitsof simple flexes.

Now one uses the automorphism σ: (x : y : z) 7→(y : z : x) toexclude the first two possibilities; moreover, one verifies that the automorphismgroup G of the curve is the semidirect product of D and < σ > (and that thesimple flexes form one G-orbit). So the degree of the orbit closure isP(d) + 3fd−3(d)3(d2 −3d + 3)= 13(d6 + 3d5 + 6d4 −21d3 −1354d2 + 5463d −5508).Next we list, for some small values of d, the numbers (and their factorizations)we get from the corollary to Theorem III:dP(d)P(d) factored321623 · 3341428023 · 3 · 5 · 7 · 17518834022 · 3 · 5 · 43 · 736111996023 · 33 · 5 · 17 · 617450828023 · 32 · 5 · 7 · 178981431825624 · 3 · 317 · 94193868074022 · 36 · 5 · 7 · 379109279048024 · 3 · 5 · 59 · 6553So for d = 3 we get 216 for the predegree of the orbit of any smooth plane cubiccurve.

This gives the well-known numbers 12, resp. 6, resp.

4 for the degree of the

LINEAR ORBITS OF SMOOTH PLANE CURVES25orbit closure of a smooth plane cubic with j ̸= 0, 1728, resp. j = 1728, resp.

j = 0.Note that the group of projective automorphisms of a smooth cubic contains the9 translations over points of order dividing 3 as a normal subgroup. The quotientcan be identified with the automorphisms that fix a given flex.

Thus there exist18, resp. 36, resp.

54 projective automorphisms when j ̸= 0, 1728, resp. j = 1728,resp.

j = 0.For d = 4 we get 14280 for the predegree of the orbit of a smooth planequartic with only simple flexes. An example of such a curve is the Klein curvex3y + y3z + z3x; it has 168 automorphisms, so the degree of its orbit closure is14280/168 = 85.If a smooth quartic has n hyperflexes (i.e., flexes of order 2), the predegreeof its orbit equals 14280 −294n.E.g., the degree of the orbit closure of theFermat quartic is 112, as there are 12 hyperflexes and 96 automorphisms.

Asan other example, consider the curve x4 + xy3 + yz3. It has 1 hyperflex and 9automorphisms, so the degree of its orbit closure is (14280−294)/9 = 1554.

In fact,in [Vermeulen] there is a complete list of the automorphism groups that occur fora quartic with a given number of hyperflexes. The implied congruence conditionsare equivalent to requiring that P(4) be divisible by 168 and that P(4) + 28f2(4)be divisible by 2016.

(This follows already from the existence of the 3 quarticsabove. )In the other direction, these formulas give non-trivial information on the au-tomorphism groups of plane curves.

Consider smooth plane curves of degree dwith only simple flexes. The least common multiple of the orders of the stabi-lizers of these curves divides P(d).

Now it is well-known that a smooth curveof positive genus g(=d−12) cannot have an automorphism of prime order p >2g + 1(= d2 −3d + 3). Using the Hurwitz formula one also excludes the cases(d, g, p) = (4, 3, 5), (6, 10, 17) and (10, 36, 59).

Looking at the table above, weconclude then that said l.c.m. divides 216, 168, 60, 1080, 2520, 48, 102060, 240respectively for d equal to 3, 4, 5, 6, 7, 8, 9, 10 respectively.These bounds seem to be pretty good: by the above, the actual l.c.m.

equals108 (resp. 168) for d = 3 (resp.

4); it’s not unreasonable to expect that the boundis sharp for d = 5, 8 and 10 (perhaps there even exist curves with automorphismgroups of this order); the Valentiner sextic has only simple flexes and 360 auto-morphisms (cf. [BHH]), so the bound for d = 6 is sharp if and only if there existsa sextic with only simple flexes and with 27 dividing the order of its stabilizer.Finally, for d = 9 the bound is probably not optimal.References[[Aluffi1]]Aluffi, P., The enumerative geometry of plane cubics I: smooth cubics, Transactions of theAMS 317 (1990), 501–539.

[[Aluffi2]]Aluffi, P., Two characteristic numbers for smooth plane curves of any degree, to appear inthe Transactions of the AMS. [[Aluffi-Faber]]Aluffi, P., Faber, C., Linear orbits of d-tuples of points in P1, preprint.

[[BHH]]Barthel, G., Hirzebruch, F., H¨ofer, Th., Geradenkonfigurationen und Algebraische Fl¨achen,Vieweg, 1987. [[Fulton]]Fulton, W., Intersection Theory, Springer Verlag, 1984.

26PAOLO ALUFFI CAREL FABER[[Vermeulen]]Vermeulen, A.M., Weierstrass points of weight two on curves of genus three, Thesis, Uni-versiteit van Amsterdam, 1983.Mathematics Department, Oklahoma State University, Stillwater OK 74078,USAMax-Planck-Institut f¨ur Mathematik, Gottfried-Claren-Straße 26, W-5300 Bonn3, Germany

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