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In this paper we extend the well known theorem of Angelo Lopez concerning the Picard group of the general space projective surface containing a given smooth projective curve, to the intermediate Neron-Severi group of a general hypersurface in any smooth projective variety.
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Let $Z$ be a closed subscheme of a smooth complex projective complete intersection variety $Ysubseteq Ps^N$, with $dim Y=2r+1geq 3$. We describe the Neron-Severi group $NS_r(X)$ of a general smooth hypersurface $Xsubset Y$ of sufficiently large degree containing $Z$.
For a binary quartic form $phi$ without multiple factors, we classify the quartic K3 surfaces $phi(x,y)=phi(z,t)$ whose Neron-Severi group is (rationally) generated by lines. For generic binary forms $phi$, $psi$ of prime degree without multiple factors, we prove that the Neron-Severi group of the surface $phi(x,y)=psi(z,t)$ is rationally generated by lines.
We investigate the global variation of moduli of linear sections of a general hypersurface. We prove a generic Torelli result for a large proportion of cases, and we obtain a complete picture of the global variation of moduli of line slices of a general hypersurface.
Let ${P_i}_{1 leq i leq r}$ and ${Q_i}_{1 leq i leq r}$ be two collections of Brauer Severi surfaces (resp. conics) over a field $k$. We show that the subgroup generated by the $P_is$ in $Br(k)$ is the same as the subgroup generated by the $Q_is$ iff $Pi P_i $ is birational to $Pi Q_i$. Moreover in this case $Pi P_i$ and $Pi Q_i$ represent the same class in $M(k)$, the Grothendieck ring of $k$-varieties. The converse holds if $char(k)=0$. Some of the above implications also hold over a general noetherian base scheme.
It is shown that an irreducible cubic hypersurface with nonzero Hessian and smooth singular locus is the secant variety of a Severi variety if and only if its Lie algebra of infinitesimal linear automorphisms admits a nonzero prolongation.
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