We make precise the structure of the first two reduction morphisms associated with codimension two non-singular subvarieties of non-singular quadrics $\Q^n$, $n\geq 5$. We give a coarse classification of the same class of subvarieties when they are assumed not to be of log-general-type.}

Keywords:

Adjunction Theory, classification, codimension two, conic bundles, low codimension, non log-general-type, quadric, reduction, special, variety. Adjunction Theory, classification, codimension two, conic bundles, low codimension, non log-general-type, quadric, reduction, special, variety.

MSC Classifications:

14C05, 14E05, 14E25, 14E30, 14E35, 14J10 show english descriptions Parametrization (Chow and Hilbert schemes)
Rational and birational maps
Embeddings
Minimal model program (Mori theory, extremal rays)
unknown classification 14E35
Families, moduli, classification: algebraic theory 14C05 - Parametrization (Chow and Hilbert schemes)
14E05 - Rational and birational maps
14E25 - Embeddings
14E30 - Minimal model program (Mori theory, extremal rays)
14E35 - unknown classification 14E35
14J10 - Families, moduli, classification: algebraic theory

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