The relations among invariants of points on the projective line

We consider the ring of invariants of n points on the projective line. The space (P^1)^n // PGL_2 is perhaps the first nontrivial example of a Geometry Invariant Theory quotient. The construction depends on the weighting of the n points. Kempe discovered a beautiful set of generators (at least in the case of unit weights) in 1894. We describe the full ideal of relations for all possible weightings. In some sense, there is only one equation, which is quadric except for the classical case of the Segre cubic primal, for n=6 and weight 1^6. The cases of up to 6 points are long known to relate to beautiful familiar geometry. The case of 8 points turns out to be richer still.Comment: 6 page announcemen

Automorphisms of classical geometries in the sense of Klein

In this note, we compute the group of automorphisms of Projective, Affine and Euclidean Geometries in the sense of Klein. As an application, we give a simple construction of the outer automorphism of S_6.Comment: 8 page

Coble fourfold, S6S_6-invariant quartic threefolds, and Wiman-Edge sextics

We construct two small resolutions of singularities of the Coble fourfold (the double cover of the four-dimensional projective space branched over the Igusa quartic). We use them to show that all $S_6$-invariant three-dimensional quartics are birational to conic bundles over the quintic del Pezzo surface with the discriminant curves from the Wiman-Edge pencil. As an application, we check that $S_6$-invariant three-dimensional quartics are unirational, obtain new proofs of rationality of four special quartics among them and irrationality of the others, and describe their Weil divisor class groups as $S_6$-representations.Comment: 57 pages; v2: minor changes; v3: referee's comments taken into account; v4: published versio

Geometry of the Wiman Pencil, I: Algebro-Geometric Aspects

In 1981 W.L. Edge discovered and studied a pencil $\mathcal{C}$ of highly symmetric genus $6$ projective curves with remarkable properties. Edge's work was based on an 1895 paper of A. Wiman. Both papers were written in the satisfying style of 19th century algebraic geometry. In this paper and its sequel [FL], we consider $\mathcal{C}$ from a more modern, conceptual perspective, whereby explicit equations are reincarnated as geometric objects.Comment: Minor revisions. Now 49 pages, 4 figures. To appear in European Journal of Mathematics, special issue in memory of W.L. Edg

Geometry of the Wiman Pencil, I: Algebro-Geometric Aspects

In 1981 W.L. Edge discovered and studied a pencil $\mathcal{C}$ of highly symmetric genus $6$ projective curves with remarkable properties. Edge's work was based on an 1895 paper of A. Wiman. Both papers were written in the satisfying style of 19th century algebraic geometry. In this paper and its sequel [FL], we consider $\mathcal{C}$ from a more modern, conceptual perspective, whereby explicit equations are reincarnated as geometric objects.Comment: Minor revisions. Now 49 pages, 4 figures. To appear in European Journal of Mathematics, special issue in memory of W.L. Edg

The geometry of eight points in projective space: Representation theory, Lie theory, dualities

This paper deals with the geometry of the space (GIT quotient) M_8 of 8 points in P^1, and the Gale-quotient N'_8 of the GIT quotient of 8 points in P^3. The space M_8 comes with a natural embedding in P^{13}, or more precisely, the projectivization of the S_8-representation V_{4,4}. There is a single S_8-skew cubic C in P^{13}. The fact that M_8 lies on the skew cubic C is a consequence of Thomae's formula for hyperelliptic curves, but more is true: M_8 is the singular locus of C. These constructions yield the free resolution of M_8, and are used in the determination of the "single" equation cutting out the GIT quotient of n points in P^1 in general. The space N'_8 comes with a natural embedding in P^{13}, or more precisely, PV_{2,2,2,2}. There is a single skew quintic Q containing N'_8, and N'_8 is the singular locus of the skew quintic Q. The skew cubic C and skew quintic Q are projectively dual. (In particular, they are surprisingly singular, in the sense of having a dual of remarkably low degree.) The divisor on the skew cubic blown down by the dual map is the secant variety Sec(M_8), and the contraction Sec(M_8) - - > N'_8 factors through N_8 via the space of 8 points on a quadric surface. We conjecture that the divisor on the skew quintic blown down by the dual map is the quadrisecant variety of N'_8 (the closure of the union of quadrisecant *lines*), and that the quintic Q is the trisecant variety. The resulting picture extends the classical duality in the 6-point case between the Segre cubic threefold and the Igusa quartic threefold. We note that there are a number of geometrically natural varieties that are (related to) the singular loci of remarkably singular cubic hypersurfaces. Some of the content of this paper appeared in arXiv/0809.1233.Comment: 31 pages, 4 figure