Weak approximation for linear systems of quadrics

Im, Bo‐Hae; Larsen, Michael

doi:10.1016/j.jnt.2005.10.002

Cited by 1 publication

(4 citation statements)

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“…Then V 0 ∩ V + w and V 0 ∩ V − w both have dimension greater than or equal to m − d, and w is positive definite and negative definite, respectively, on them. 2 According to [4,Proposition 4], if the k-dimensional space of quadratic forms on V is (k 2 − k + 1)-admissible, the (non-linear) evaluation map E : V → W * given by…”

Section: Theoremmentioning

confidence: 99%

“…the number of mutually orthogonal hyperbolic planes in the inner product space defined by Q, is at least r + 1. In this paper we give a qualitative generalization of this result to intersections of quadric hypersurfaces.Such intersections were considered from a rather different point of view in [4]. That paper proved weak approximation for rational points on PY W , the set of simultaneous solutions of quadric hypersurfaces indexed by W , under an admissibility condition given in Definition 1 below.…”

mentioning

confidence: 99%

“…Such intersections were considered from a rather different point of view in [4]. That paper proved weak approximation for rational points on PY W , the set of simultaneous solutions of quadric hypersurfaces indexed by W , under an admissibility condition given in Definition 1 below.…”

mentioning

confidence: 99%

“…According to [IL,Prop. 4], if the k-dimensional space of quadratic forms on V is (k 2 − k + 1)-admissible, the (nonlinear) evaluation map E : V → W * given by ( 1)…”

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confidence: 99%

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The higher connectivity of intersections of real quadrics

Larsen

Lindenstrauss

2006

Topology and its Applications

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A linear system of real quadratic forms defines a real projective variety. The real non-singular locus of this variety (more precisely of the underlying scheme) has a highly connected double cover as long as each non-zero form in the system has sufficiently high Witt index.There is an extensive literature (see, e.g., [1] and the references contained therein) concerning connectivity theorems for complex projective varieties. The philosophy, ever since Lefschetz proved his hyperplane theorem, has been that complex projective subvarieties of low codimension should have the same low-dimensional homology and homotopy groups as their ambient varieties. The situation for real algebraic varieties is more delicate. Even in the simplest non-trivial case, non-singular quadric hypersurfaces in RP n , one cannot make a connectedness statement without further hypotheses. For example, the quadric Q = x 2 0 + · · · + x 2 m−1 − x 2 m − · · · − x 2 n = 0 in RP n does not necessarily look like RP n in low dimensions: If we pull it back by the double cover S n → RP n , we obtain a space homeomorphic to S m−1 × S n−m which is r-connected if and only if the Witt index of Q, i.e. the number of mutually orthogonal hyperbolic planes in the inner product space defined by Q, is at least r + 1. In this paper we give a qualitative generalization of this result to intersections of quadric hypersurfaces.Such intersections were considered from a rather different point of view in [4]. That paper proved weak approximation for rational points on PY W , the set of simultaneous solutions of quadric hypersurfaces indexed by W , under an admissibility condition given in Definition 1 below. Admissibility guarantees that many lines in the ambient projective space lie in PY W . In fact, for any two points P , Q ∈ PY W , there exists R ∈ PY W such that P R and QR lie in PY W .

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Section: Theoremmentioning

confidence: 99%

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confidence: 99%

mentioning

confidence: 99%

“…According to [IL,Prop. 4], if the k-dimensional space of quadratic forms on V is (k 2 − k + 1)-admissible, the (nonlinear) evaluation map E : V → W * given by ( 1)…”

mentioning

confidence: 99%

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