Vol. 46, No. 4 DUKE MATHEMATICAL JOURNAL (C) December 1979 GALOIS GROUPS OF ENUMERATIVE PROBLEMS JOE HARRIS Contents. Oo Introduction 685 I. Galois and monodromy groups 688 II. Flexes and bitangents of plane curves 690 1. Pliicker formulas 690 2. Flexes of a plane cubic 693 3. Flexes of a plane curve of degree d > 4 697 4. Bitangents to a smooth plane quartic 700 5. Bitangents of a plane curve, degree d > 5 706 III. Lines on hypersurfaces 707 1. The Fano scheme 707 2. Lines on a hypersurface in P", n > 4 712 3. Lines on a cubic surface 715 4. Bitangents of a plane quartic II 719 IV. The problem of five conics 721 References 724 0. Introduction. This paper is concerned with the solvability of certain enumerative problems in algebraic geometry. To illustrate the questions raised, consider one of the problems dealt with belowmthat of locating the flexes of a plane curve. We let C c P be a complex plane curve of degree d, given as the locus of the homogeneous polynomial F(Xo,X,Xa)= ZagjX-i-Jx[xJ2 (or in j. euclidean coordinates x Xi/Xo, as the locus of f(x,x2) ,aijxlxa), we will take coefficients ai. to be general complex numbers. At a generic point of C, then, the tangent lne Tp(C) intersects C with multiplicity mp(C. I)= 2; we say that p is a flex point of C if mp(l. C)> 3. An elementary count of parameters leads us to expect that C will have a finite number of flex points p; and accordingly we may ask two questions: first, how many flexes does C possess? and second, can we find them?--that is, is it possible to give a formula for coordinates xi(p) of the flex points p of C in terms of the coefficients There are a number of ways of answering the first of these questions; historically the first was to note that the flexes of C comprise the locus F(X) H(X)= O, where H(X) is the Hessian det H(X) OXiOXj Received February 13, 1979. Research partially supported by NSF Grant MCS 77-01964. 685 686 JOE HARRIS since H(X) has degree 3(d-2), we expect 3d(d-2) flexes. To set up the second problem, we note that since the coordinates x(p,) are roots of the ith resultant Ri(f, h), (xi)a polynomial of degree 3d(d- 2) in x whose coefficients are elements of the field G(aj) of rational functions in the ao.the field C,(ai,x(p,O) obtained from G(a0.) by adjoining the xi(p,0 is a finite algebraic extension of C(aj); and there will exist a formula for xi(p,,) in terms of the a/j exactly when the Galois group G of this extension is solvable. We may thus replace our second question with the broader, What is the Galois group G of the extension C,(ai,xi(pa))/C,(aij) 9. We note that determining G will also answer some other questions in the same vein. For example, knowing G and the action of G on the roots of Ri, we may ascertain whether the subgroup G’ GalG(ao.,xi(p))/G(aij,xi(Pl),..., xi(pk)) of G fixing k roots is solvable-- that is whether, given the coordinates of k flexes of C, we may solve explicitly for the remaining 3d(d- 2)- k. In similar fashion, there is associated to most enumerative problems a Galois group, whose structure reflects the difficulty of obtaining explicit solutions to the problem. Historically, these Galois groups were first studied systematically by Camille Jordan in his book Traitb des Substitutions (1870) where he discusses, among others, the Galois groups of the flexes of a plane cubic, the bitangents to a plane quartic curve, and the lines on a cubic surface in [a3. Jordan’s work, however--as well as later refinements by Dickson [4] and Weber [13]--was in a direction opposite to that suggested above: the equations defining the coordinates of the flexes of a plane cubic, bitangents to a plane quartic and lines in a cubic surface had already been extensively studied by Cayley, Clebsch, Hesse, Steiner and others ([2], [3], [12]) some time before the advent of Galois theory. What Jordan did was to show how this earlier work would be expressed and interpreted in terms of Galois theory. Thus, for example, in considering the flexes of a plane cubic, Jordan takes Hesse’s solution of the problem and analyzes it to determine the group. In the present paper we take the opposite tack, and determine the solvability of four enumerative problems--the first three of which are generalizations of the ones mentioned above--by computing their Galois groups. There are, in general, two steps in solving each of these problems: First, we look to see what, if any, intrinsic structure is carried by the set of solutions to the problem (e.g., among the nine flexes of a plane cubic, there are 12 sets of three collinear points). This structure must be preserved under the action of the Galois group G (e.g., if Pl, P2, P3 are three collinear flexes of a plane cubic F(X)= 0, the coordinates of P3 may be rationally expressed in terms of those of P and P2 by restricting F to the line PiP2 and factoring out the roots corresponding to _Pl and P2; and so must be preserved under any automorphism 0 G fixing xi(Pl) and xg(p2)). This then gives us an "upper bound" on the group. The second half of the problem--showing that G must be at least as large as some subgroup H of the symmetric group on the set of solutionsseems harder. Certainly it is not practical to analyze the equation in generalfor d large one GALOIS GROUPS OF ENUMERATIVE PROBLEMS 687 could hardly hope, for example, to determine whether the group G of the flexes of a plane d- ic was contained in the alternating group by writing down the discriminant of the degree 3d(d- 2) polynomial R and determining whether it was a square in G(aj). Our technique, rather, is to equate the group G with the monodromy group M of a generically finite map of algebraic varieties, as described in chapter of this paper. This allows us to exhibit elements of the Galois/monodromy group simply by drawing arcs in the parameter space for the problem. One situation in which this approach is particularly effective is in showing that G is the full symmetric group. Here, after showing that G is twice transitive, we want to exhibit a simple transposition in G; and by the lemma of section 11.3 below, this amounts--modulo some secondary argumentsmto locating a special case of the problem which admits a "solution of multiplicity tWO." The four problems considered here are the flexes and bitangents of a general plane curve of any degree d; the lines on a general hypersurface 1/" c an of degree 2n 3, for any n; and the conics tangent to each of five given conics in the plane. For each of the first three, we consider separately the case studied by Jordan, giving a new proofmusing monodromy argumentsmof his theorem, and say what this means in terms of solutions to the problem. We then consider the general case. The arguments in all four cases follow pretty much the same lines; as indicated above, the essential ingredient is the determination of the multiplicity of a solution. For the flexes and bitangents of plane curves, this is implicit in the general form of the classical Pliicker formulas; in the case of lines in hypersurfaces it is a straightforward computation made in section III.1; and for the conics tangent to five conics the answer was recently supplied by Fulton and MacPherson in [5]. The findings may be stated readily: the Galois groups are (i) For the flexes of a plane d- ic, the affine special linear group ASL2(Z/3) in case d 3; and the full symmetric group in case d > 4; (ii) For the bitangents to a plane d- ic, the orthogonal group O6(Z/2) in case d 4 and the full symmetric group in case d > 5; (iii) For the lines on a hypersurface l/’2n_31a: the odd orthogonal group 06- (Z/2) in case n 3, and the full symmetric group in case n > 4; and (iv) For the conics tangent to five conics, the full symmetric group. One remark, finally, about the significance of the results, and in particular the fact that, in every case not considered classically, the group is shown to be the full symmetric group. On one hand, it is of course a disappointment that no explicit solutions may be given for the problems mentioned. On the other hand, however, the results represent an affirmation of one understanding of the geometry underlying each of these problems, in the following sense: in every case dealt with here, the actual structure on the set of solutions of the enumerative problem as determined by the Galois group of the problems, is readily described in terms of standard algebrao-geometric constructions. In particular, in every case in which current theory had failed to discern any 688 JO HARRIS intrinsic structure on the set of solutionsit is proved herethere is in fact none. I would like to thank Steven Kleiman, who introduced this problem to me, and whose ideas have guided this work throughout.