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Projective Plane Curves Whose Automorphism Groups Are Simple and Primitive

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Projective Plane Curves Whose Automorphism Groups Are Simple and Primitive Projective plane curves whose automorphism groups are simple and primitive Yusuke Yoshida Abstract We study complex projective plane curves with a given group of auto- morphisms. Let G be a simple primitive subgroup of PGL(3, C), which is isomorphic to A6, A5 or PSL(2, F7). We obtain a necessary and sufficient condition on d for the existence of a nonsingular projective plane curve of degree d invariant under G. We also study an analogous problem on integral curves. 1 Introduction Automorphism groups of algebraic curves have long been studied. For example, Hurwitz found an upper bound on the order of the automorphism group of a curve of a given genus ([Hurwitz]). In the case of plane curves, there are more detailed studies on automorphism groups. One important fact here is that an automorphism of a smooth projective plane curve of degree greater than or equal to 4 uniquely extends to an automorphism of the projective plane. Hence, the automorphism group of such a curve is isomorphic to a subgroup of PGL(3, C). Recently, Harui obtained the following result concerning the classification of automorphism groups of smooth plane curves. Theorem 1.1. [Harui, Theorem 2.3] Let C be a smooth plane curve of degree d 4, and G a subgroup of Aut C. Then one of the following holds: ≥ (a-i) G fixes a point on C, and G is cyclic. (a-ii) G fixes a point not lying on C, and up to conjugation there is a commu- tative diagram π 1 / C× / PBD(2, 1) / PGL(2, C) / 1 arXiv:2008.13427v1 [math.AG] 31 Aug 2020 S S S 1 / N / G / G′ / 1 with exact rows where the subgroup PBD(2, 1) of PGL(3, C) is defined by A O PBD(2, 1) := PGL(3, C) A GL(2, C), α C× , O α ∈ ∈ ∈ N is cyclic and G′ is isomorphic to a cyclic group Z/mZ, a dihedral group m, the tetrahedral group A , the octahedral group S or the icosahedral D2 4 4 group A5. 1 d d d (b-i) (C, G) is a descendant of the Fermat curve Fd defined by x + y + z = 0. (For the definition of a descendant, see [Harui, Definition 2.2].) In particular, G is conjugate to a subgroup of Aut Fd. d−1 d−1 (b-ii) (C, G) is a descendant of the Klein curve Kd defined by x y + y z + zd−1x = 0. In particular, G is conjugate to a subgroup of Aut Kd. (c) G is a finite primitive subgroup of PGL(3, C). (For the definition of a primitive subgroup of PGL(3, C) and more details, see Definition 2.5.) In this case, G is conjugate to one of the following groups: The Valentiner group , which is isomorphic to A . • V 6 The icosahedral group , which is isomorphic to A . • I 5 The Klein group , which is isomorphic to PSL(2, F ). • K 7 The Hessian group H216 of order 216 or its subgroup of order 36 or • 72. There are also more concrete studies of the automorphism groups. For ex- ample, the list of all automorphism groups of projective plane curves of degree 5 was given by E. Badr and F. Bars ([BB]). Beyond that, it seems that we still do not have the list of the automorphism groups of projective plane curves of a given degree. We look at the problem from a different point of view. Fix a group G in the classification, consider nonsingular (resp. integral) projective plane curves C invariant under G. Then we want to find all possible values of the degree of such a curve C. For a group G of type (a-i) or (a-ii) in the classification, invariant curves under G are related with the study of Galois points and are actively studied. (See e.g. [HMO].) For a pair (C, G) as in (b-i) or (b-ii) in the classification, C is defined by d d d i j k x + y + z + cijkx y z =0 i+j+k=d,Xmax{i,j,k}<d or d−1 d−1 d−1 i j k x y + y z + z x + cijkx y z =0. i+j+k=d,maxX{i,j,k}<d−1 and G acts on C. Thus it can be said that we have a good understanding of these cases. In this paper, we focus on groups of type (c) in the classification and study invariant curves under each of them except for subgroups of H216 and its sub groups. As the first main result, we determine all degrees of nonsingular pro- jective plane curves invariant under each of the groups = A , = A and V ∼ 6 I ∼ 5 = PSL(2, F ): K ∼ 7 Theorem 1.2. Let d be a positive integer. (1) There exists a nonsingular projective plane curve of degree d whose auto- morphism group is equal to if and only if d 0, 6 or 12 mod 30. V ≡ 2 (2) There exists a nonsingular projective plane curve of degree d invariant under if and only if d 0, 2 or 6 mod 10. I ≡ (3) There exists a nonsingular projective plane curve of degree d whose auto- morphism group is equal to if and only if d 0, 4 or 6 mod 14. K ≡ We note that if C is invariant under or then Aut C is equal to or , respectively. V K V K Next, we give a result on integral (i.e. irreducible and reduced) curves in- variant under each of the groups above. We again find all degrees of such curves: Theorem 1.3. Let d be a positive integer. (1) There exists an integral projective plane curve of degree d invariant under if and only if d is a multiple of 6, d = 18 and d = 24. V 6 6 (2) There exists an integral projective plane curve of degree d invariant under if and only if d is even and is neither 4, 8 nor 14. I (3) There exists an integral projective plane curve of degree d invariant under if and only if d is even and is neither 2, 8, 12, 16 nor 22. K We prove these results in the following way. The groups , and have been classically well studied as reflection groups. (Some resultsV areI summarizedK in [ST], for example.) If G is a simple primitive finite subgroup of PGL(3, C), which is conjugate to , or , then we can take a lift G of G in GL(3, C) such V I K that a curve invariant under G is defined by a polynomial invariant under G. In e addition, the invariant rings were studied for A in [Wiman] and for PSL(2, F ) 6 e 7 in [Klein] around the end of the 19th century. By regarding A5 as a subgroup of A6, we can give the invariant ring of A5. For a given degree d, we can thus describe the linear system (dG)d of G- invariant curves of degree d using explicitly given invariant polynomials FG,ΦG and ΨG. We can translate conditions on the base locus of (dG)d to congruence relations on d. Then we look at singularities of a general element of (dG)d with the help of Bertini’s theorem and study when it is nonsingular at the base points. We next consider integral curves. The problem is to show that there exist integral invariant curves of degree d in the theorem. In this case, it can be shown that a general element C is reduced in the same way. Since any nonsingular plane curve is integral, it suffices to consider degrees d for which (dG)d has only singular elements. It turns out that the singular points form a G-orbit, hence are of the same type. We can also give the number of singularities and the type of singularities of a general member. From this information, we see, if a general element were reducible, it would have too much singularity. The organization of this paper is as follows. In Section 2, we recall a number of facts needed for the discussion of invariant curves. We give a description of homogeneous polynomials invariant under the group , or following [Crass] and [Elkies]. In Section 3, we first give a conditionV I for theK existence of a nonsingular invariant curve in a form that is valid for any of , and , and then translate these conditions for each group. In Section 4,V weI study integralK invariant curves. We look at singularities of a general invariant curve of a degree where there is no nonsingular invariant curve and prove that it cannot 3 be reducible. In Appendix, we give some codes in SINGULAR to check a few calculations in Section 2. 2 Simple primitive finite subgroups of PGL(3, C) and invariant curves In this section, we give a description of the groups , and , their invariant rings and the invariant curves. V I K First, we give basic definitions related to invariant homogeneous polynomials. Notation. Let x be the coordinate (x,y,z) in C3. We often identify x with x the row vector (xyz) or the column vector y . z Definition 2.1. (1) For a matrix A GL(3, C) and a homogeneous polyno- mial f(x,y,z) C[x,y,z], we set ∈ ∈ (f A)(x,y,z) := f xtA = f (Ax) . Note that this is an action from the right. (2) Let G be a subgroup of GL(3, C).
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