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Galois Groups of Enumerative Problems

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Galois Groups of Enumerative Problems Galois groups of enumerative problems Yongquan Zhang Advisor: Prof. Joe Harris Minor thesis Department of Mathematics Harvard University January 2019 Contents 1 Introduction 2 2 Preliminaries 3 3 Galois groups of flexes of plane curves 11 4 Galois groups of bitangents of plane curves 15 5 Galois group of 27 lines on a cubic surface 19 6 Galois group of the problem of five conics 26 7 Resolvent degree 30 References 33 1 1 Introduction In this report, we discuss the solvability of some enumerative problems following [Har1]. To illustrate the type of problems at hand, take the famous example of 27 lines on a cubic surface 3 3 in P . Let S ⊂ P be a cubic surface given as the locus of a degree 3 homogenous polynomial P I 3 F (X) = #I=3 aI X in some homogenous coordinate X = [X0 : X1 : X2 : X3] on P . A line 3 in P is given by the span of two linearly independent vectors, say A = (a0; a1; a2; a3) and B = (b0; b1; b2; b3); for a generic cubic surface, we may even assume a0 = 0 = b1; a1 = 1 = b0. 1 Then the line lies on F (X) if and only if F (sA + tB) = 0 for all [s : t] 2 P . This gives 4 degree 3 (nonhomogenous) polynomials in a2; a3; b2; b3, with coefficients as polynomials in aI 's. Hence we do expect finitely many lines lying on the cubic surface. The question is then how many, and given coefficients aI , whether we can solve for these lines algebraically. To interpret solvability in terms of Galois theory, note that the solutions to a system of polynomial equations have algebraic coordinates over the base field, which is, in our case above, the field of rational functions C(aI ) in the coefficients aI . In particular, one may ask about the solvability of the Galois group of the algebraic extension obtained by adjoining the coordinates of all the solutions to C(aI ). More generally, one may ask about the solvability of this algebraic extension over C(aI ) adjoining the coordinates of some of the solutions, which in our case is equivalent to ask if we can solve for the remaining lines given the coordinates of some of them. Note that C(aI ) is essentially the function field of the complete linear system of cubic 3 surfaces in P . Consider the incidence correspondence in the space of pairs of a cubic surface and a line, cut out by the condition that the line lies on the cubic surface. Clearly the algebraic extension mentioned in the previous paragraph gives, in essence, the function field of this algebraic variety. There is also a natural projection from the incidence correspondence to the complete linear system. This projection is of finite degree (27 in this case), and hence we can define the associated monodromy group (for details, see Section 2.6). The key observation is that the Galois group and the monodromy group coincide, and hence we may study the Galois group by looking at the monodromy group and the rich geometry there. The Galois group then reveals information about the solutions; in particular, determining the solvability when the coordinates of some of the lines are given is the same as determining the solvability of the corresponding subgroup. This approach enables us to study this type of problems systematically. Instead of look- ing at solutions to find additional structures, we look at the group directly. This is in direct contrast to more classical explorations of these problems, where one obtain solvability by study the structure of the solutions, and then determine the Galois group from the informa- tion. For example, the Galois group acts on the 27 lines on the cubic surface preserving the intersection configuration, and hence cannot be the full symmetry group. We will comment on this classical approach when we can, drawing from references such as [Jor]. The structure of this report is as follows: some preliminary facts are reviewed in Section 2, many of which come from a basic course in algebraic geometry, mostly about plane curves, and are included for the benefit of the author, whose knowledge in the subject has been somewhat 2 rusty. The remaining sections are then each devoted to a different enumerative problem. Our focus will be the problems of flexes and bitangents on plane curves. In order to solve the problem of bitangents on quartics completely, we also need the solution to the problem of 27 lines on a cubic surface. After that, we include a short exposition on the problem of five conics to showcase the variety of problems that may be settled using this approach. Finally, we give an account of resolvent degree, including some more recent related research. 2 Preliminaries Some references for this section include [Har2, Ful, GH]. 2.1 Flexes and bitangents of plane curves A plane curve of degree d is the vanishing locus of a homogeneous polynomial F (X0;X1;X2) 2 of degree d in P . The (arithmetic) genus of a plane curve is given by g = (d − 1)(d − 2)=2 by the genus-degree formula. 2 Fix some homogenous coordinate [X0 : X1 : X2] on P , and corresponding dual coordinate 2∗ 2 2 [Y0 : Y1 : Y2] on P , the projective space of lines in P . That is, a point [X0 : X1 : X2] on P P lies on a line corresponding to [Y0 : Y1 : Y2] if and only if i XiYi = 0. Given a smooth plane 2∗ curve C of degree d, we can define a map f : C ! P by sending p 2 C to the tangent line of C at p. This map is algebraic: indeed, it is the restriction of the map " # @F @F @F p = [X0 : X1 : X2] 7! : : @X0 p @X1 p @X2 p to the curve C. Thus the image of C is also a curve, called the dual curve and denoted by C∗. Moreover the map f is generically one-to-one, as easily verified from its expression given above. The degree d∗ of C∗ is readily determined: it is the number of intersection of a generic 2∗ ∗ 2∗ 2 line in P with C . A line in P corresponds to a point in P , and thus the degree is also 2 the number of tangent lines to C passing through a generic point in P . Given a point q, assume none of the lines passing through q is multiply tangent to C. Consider the projection of C from q onto a generic line. This is a map of degree d, ramified at points of tangency. By Riemann-Hurwitz, the number of branch points is given by 2g − 2 + 2d = (d − 1)(d − 2) − 2 + 2d = d(d − 1): P This is as expected, as these tangent points are solutions of F (X) = i ai@F=@Xi = 0 where ∗ ∗ [a0 : a1 : a2] are the coordinate of q. Hence the degree d of the curve C is d(d − 1). The discrepancy in degree of C and C∗ is accounted for by the fact that C∗ is not smooth even if C is. We now try to determine the singularities of C∗. Let p 2 C be a point such that 3 the tangent line to C at p has contact order m ≥ 2. For simplicity, we choose homogeneous 2 coordinates on P and (holomorphic) local coordinate t on C, so that p = [1 : 0 : 0] and C is parametrically given locally (in an analytic small neighborhood) as t 7! [1 : t : X2(t)] m where X2(t) vanishes at t = 0 to degree m. Write X2(t) = t v(t) for some nonzero holomor- phic function v in a neighborhood of zero. Then one can calculate, locally, the expression of f: 0 0 m m+1 0 m−1 m 0 f(t) = [tX2(t)−X2(t): −X2(t) : 1] = (m − 1)t v(t) + t v (t): −mt v(t) − t v (t) : 1 m m−1 Therefore, in some local coordinate et, f is given as t~ 7! [t~ u(t~): t~ : 1] for some nonzero holomorphic function u in a neighborhood of zero. In particular, we see that C∗ has a cuspidal singularity of multiplicity m − 1 at f(p) and the map f is smooth at p if and only if m = 2. Moreover, from the expression for f, which gives a parametrization of C∗ locally, we see that at point f(p) of C∗, the tangent line to C∗ corresponds to p. As we have discussed, the map f is generically one-to-one over a point l 2 C∗, unless the ∗ corresponding line l is tangent to C at several points p1; : : : ; pe. For each branch of C at l, ∗ the tangents are distinct, as these tangents correspond to points pi respectively. Hence C has a e-fold ordinary singularity at l. We will always assume there are only two types of singularity at a point l 2 C∗: ordinary double point and cusps of degree 2. In particular, these correspond to lines tangent to C at two distinct points, each with contact order 2, and lines tangent to C at a unique point with contact order 3. The former will be referred to as bitangents, and the latter ordinary flexes (or simply flexes when no confusion arises; in general, a flex is a point at which the tangent line has contact order at least 3). This assumption holds for generic curves of degree 2 d. Indeed, let Wd be the complete linear system of plane curves of degree d in P .
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