M-curves of degree 9 with deep nests

The first part of Hilbert's sixteenth problem deals with the classification of the isotopy types realizable by real plane algebraic curves of given degree $m$. For $m \geq 8$, one restricts the study to the case of the $M$-curves. For $m=9$, the classification is still wide open. We say that an $M$-curve of degree 9 has a deep nest if it has a nest of depth 3. In the present paper, we prohibit 10 isotopy types with deep nests and no outer ovals.Comment: 16 pages, 11 figures v.4 minimal correction

Apparent contours of nonsingular real cubic surfaces

We give a complete deformation classification of real Zariski sextics, that is of generic apparent contours of nonsingular real cubic surfaces. As a by-product, we observe a certain "reversion" duality in the set of deformation classes of these sextics.Comment: 61 pages, 8 figures, Revised version to be published in Transactions AMS: some minor corrections, a missing lemma is include

Deformation classification of real non-singular cubic threefolds with a marked line

We prove that the space of pairs $(X,l)$ formed by a real non-singular cubic hypersurface $X\subset P^4$ with a real line $l\subset X$ has 18 connected components and give for them several quite explicit interpretations. The first one relates these components to the orbits of the monodromy action on the set of connected components of the Fano surface $F_\mathbb{R}(X)$ formed by real lines on $X$. For another interpretation we associate with each of the 18 components a well defined real deformation class of real non-singular plane quintic curves and show that this deformation class together with the real deformation class of $X$ characterizes completely the component

Topological types of real regular jacobian elliptic surfaces

We present the topological classification of real parts of real regular elliptic surfaces with a real section.Comment: 17 pages, 7 figures, to appear in Geometriae Dedicat