Taarab Music in Zanzibar in the Twentieth Century

Janet Topp Fargion

2016

Abstract

It is known that the dynamical degree of an automorphism g of an algebraic surface S is lower semi-continuous when (S, g) varies in an algebraic family. In this paper we report on computer experiments confirming this behavior with the aim to realize small Salem numbers as the dynamical degrees of automorphisms of Enriques surfaces or rational Coble surfaces.

Figures (9)

Figure 1: The Enriques lattice Ej9 and the Coxeter diagram of its reflection group Its Coxeter diagram is of T-shaped type

The deck transformation o of f extends to a biregular automorphism of X. It acts on the components of C as follows

The results of the computations of the dynamical degrees are given in the following table.

4 Enriques surfaces of Hessian type Here we assume that char(k) 4 2,3. 1.1 Cubic surfaces and their Hessian quartic surfaces

Figure 2 is the picture of the Sylvester pentahedron with vertices P,, and edges Lap Each point P,» is contained in three lines L,g. The lines and points form a well-known abstract Desargues configuration (103) (see [15], III, §19). The birational transformation

Table 1: Salem numbers for a general Enriques surface of Hesse type We will restrict ourselves with elements g € G which are products of N different hap’s. The support of such a word in the generators defines a subgraph of the Petersen graph of cardinality N. Different graphs may define different conjugacy classes and the same subgraph may correspond to different conjugacy classes. The following list gives the smallest Salem numbers which we were able to find in this way. °

As before, we have involutions hay defined by the projections from the nodes. It follows from Corollary 4.2 that h7,,a < b < 4, coincides with the transposition t,,. Computing the action of products of different involutions, we can substantially decrease the spectral radii. Our best results are given in Table 2. Table 2: Salem numbers for a special pencil of Enriques surface of Hesse type Here we order the pairs as (12,13, 14, 15, 23, 24, 25,34,35,45) and give the sequence of subscripts in the product of the hay’s. Note that the Salem number of degree 2 is the smallest possible and the Salem number of degree 4 is the third smallest possible. where w is the third root of 1 different from 1.

Figure 4: The image of the fiber of type Jj on Vi Assume now that t = i: In this case vi is generated over Q by the classes of the curves Uap and the classes Cp — Cy, C3 — C1, C4 = C,. Its rank is equal to 13. One checks that the hyperplane section 2g + 2» = 0,1 <a< 6b < 4, intersects the Hessian along the union of the edge rg = 2 = 0 and a line (4) = yg + 2% = Xe — Xa = 0, {a,b} 1 {k, 1} = 0 taken with multiplicity 2. The Cremona involution ®,, leaves the lines @,) invariant. The hyperplane section spanned by the line ¢,» and the edge rq = xp = 0 is tangent to H1,/4 along the edge and the line. The pencil of hyperplanes through the edge defines an elliptic fibration on X1 with two reducible fibers of type Jg, one reducible fiber of type Jj and one reducible fiber of type J4. The latter two fibers correspond to the planes 7, + rt» = 0 and x, — x = 0, respectively. Each plane contains a pair of new singular points. The fiber of type Jj is equal to the divisor D,; = 2R9 + Ri + Ro + R3 + Ry, where Ro is the proper inverse transform of the line £,,, the curves R,, Rg are the exceptional curves over the new singular points lying on £4) which are pointwisely fixed by ®, and the curves R3, R4 are the curves Ng and Tea. The image of D, on the Coble surface Vi is the divisor 4E9 + 2E, + Eo + E3, where 4 ER = -1, EF? = -2, Ef = FE} = —4 and Ey- BE, = 1,1=1,2,3,E;- Ej =0,1<i<j =3. Its image on the relatively minimal elliptic surface is a fiber of Kodaira’s type III. The fiber of type J, is a divisor Dz = A, + Ag + A3 + Ay forming a quadrangle of (—2)- curves. Two opposite sides A, Ag are the proper transforms of the components of the residual cubic curve. The curves A3, A, are the exceptional curves over two new singular points. The image of D2 in Vi is a quadrangle of four smooth rational curves as pictured on Figure 5. Its image on the relatively minimal elliptic surface is a fiber of type Io.

Figure 5: The image of the fiber of type J, on Vi