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Common Transversals and Tangents: Statement of the Theorem
Theorem(G. Megyesi, F. Sottile and Th. Theobald)

Fix 2 skew lines L1 and L2 and a general quadric Q in 3-dimensional projective space.
The closure of the set of quadrics for which there are infinitely many lines simultaneously transversal to L1 and L2 and tangent to both quadrics is a curve in the 9-dimensional projective space of quadrics. This curve consists of 12 plane conics.

Remark
The case of two lines meeting in a point reduces the question to a planar problem.

Idea of Proof:

  • Determine ideal defining the set of quadrics.
  • Identify and remove components not contributing to the answer.
  • Analyze the remaining components
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Sao Carlos, 08/02 http://www.singular.uni-kl.de