Abstract. A rigid isotopy of nonsingular real algebraic curves on a quadric is a path in the space of such curves of a given bidegree. We obtain the rigid isotopy classification of nonsingular real algebraic curves of bidegree (3,3) on a hyperboloid and on an ellipsoid. We also study the space of real algebraic curves of bidegree (3,3) with a single node or cusp. Bibliography: 11 items. , and the classification of their complex schemes (i.e., real schemes enriched with a type and complex orientations, see below), in [8]. In the present paper we prove that a nonsingular curve of bidegree (3,3) on a hyperboloid (on an ellipsoid) is determined up to rigid isotopy by its complex (respectively, real) scheme, see Theorem 2. In the proof of Theorem 1 we enumerate all the connected components of the space of curves of bidegree (3,3) with a single node or cusp (see Figures 1 and 2). We use the approach to the rigid isotopy classification of plane real quartics suggested in [9].
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