Abstract
In this paper we study intersections of quadrics, components of the hypersurface in the Grassmannian G r (3, ℂ n ) introduced by S. Sawada, S. Settepanella and S. Yamagata in 2017. This lead to an alternative statement and proof of Pappus’s Theorem retrieving Pappus’s and Hesse configurations of lines as special points in the complex projective Grassmannian. This new connection is obtained through a third purely combinatorial object, the intersection lattice of Discriminantal arrangement.
Highlights
Pappus’s hexagon Theorem, proved by Pappus of Alexandria in the fourth century A.D., began a long development in algebraic geometry.In its changing expressions one can see reflected the changing concerns of the field, from synthetic geometry to projective plane curves to Riemann surfaces to the modern development of schemes and duality.(D
We conjecture that regularity in the geometry of Discriminantal arrangement could lead to results on hyperplanes arrangements with high multiplicity intersections, e.g., in the case k = 3, line arrangements in P2 with high number of triple points
In the last section we study intersections of higher numbers of quadrics and Hesse configuration
Summary
Pappus’s hexagon Theorem, proved by Pappus of Alexandria in the fourth century A.D., began a long development in algebraic geometry. In this paper we look at Pappus’s configuration (see Figure 1) as a generic arrangement of 6 lines in P2 which intersection points satisfy certain collinearity conditions (see Figure 2) This allows us to apply results on [6] and [10] to restate and re-prove Pappus’s Theorem. In the rest of the paper, we retrieve the Hesse configuration of lines studying intersections of six quadrics of the form Qσ for opportunely chosen [σ] This lead to a better understanding of differences in the combinatorics of Discriminantal arrangement in the complex and real case. We conjecture that regularity in the geometry of Discriminantal arrangement could lead to results on hyperplanes arrangements with high multiplicity intersections, e.g., in the case k = 3, line arrangements in P2 with high number of triple points (see Remark 6.6) This will be object of further studies. In the last section we study intersections of higher numbers of quadrics and Hesse configuration
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