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Some (non-)elimination results for curves in geometric structures

Serge Randriambololona, Sergei Starchenko (2011)

Fundamenta Mathematicae

We show that the first order structure whose underlying universe is ℂ and whose basic relations are all algebraic subsets of ℂ² does not have quantifier elimination. Since an algebraic subset of ℂ² is either of dimension ≤ 1 or has a complement of dimension ≤ 1, one can restate the former result as a failure of quantifier elimination for planar complex algebraic curves. We then prove that removing the planarity hypothesis suffices to recover quantifier elimination: the structure with the universe...

Some remarks on Set-theoretic Intersection Curves in P 3

Roberto Paoletti (1996)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Motivated by the notion of Seshadri-ampleness introduced in [11], we conjecture that the genus and the degree of a smooth set-theoretic intersection C P 3 should satisfy a certain inequality. The conjecture is verified for various classes of set-theoretic complete intersections.

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