On the cohomology of the space of seven points in general linear position
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RESEARCH
On the cohomology of the space of seven points in general linear position Olof Bergvall∗ * Correspondence:
[email protected] Department of Electronics, Mathematics, and Natural Sciences, University of Gävle, Gävle, Sweden
Abstract We determine the cohomology groups of the space of seven points in general linear position in the projective plane as representations of the symmetric group on seven elements by making equivariant point counts over finite fields. We also comment on the case of eight points. Keywords: Configurations of points, Moduli spaces, Ètale cohomology, de Rham cohomology, Group actions, Representations of finite groups Mathematics Subject Classification: 14N20 (primary)14H10, 14J10, 14J45, 14F20, 14F40, 14L30 (secondary)
1 Introduction Given a variety X, it is very natural to consider m-tuples (P1 , . . . , Pm ) of points on X. If no condition is placed on the tuples, the space of these tuples is X m and the space of tuples such that the points are distinct is the configuration space Confm (X). If the order of the points is irrelevant, one instead arrives at the symmetric product S m (X) and the unordered configuration space UConfm (X), respectively. The ordered and unordered spaces are naturally related through the action of the symmetric group Sm permuting the points. It is both possible and interesting to pose more refined requirements on the m-tuples; For instance, tuples of points in general position, i.e., such that there is no “unexpected” subvariety containing the points, have been studied extensively in the classical literature. See the book [9] of Dolgachev and Ortland for a modern account on this topic and further references. Tuples of points in general position also have close connections to various moduli spaces of curves, surfaces, and abelian varieties; see, e.g., [1,3,19]. We also mention that spaces of m-tuples in general position in P2 have been studied from a cohomological point of view by Gounelas and the author when m is at most seven; see [3–5]. The techniques employ both point counts over finite fields and purity arguments, two topics that are discussed further in the present paper. In Pn , it is also natural to consider m-tuples of points in general linear position. In other words, we require that no subset of n + 1 points should lie on a hyperplane. We denote the glp space of m-tuples of points in Pn which are in general linear position by Confm (Pn ) and
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