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Let G be a group and let V be an algebraic variety over an algebraically closed field K. Let A denote the set of K-points of V. We introduce algebraic sofic subshifts 
${\Sigma \subset A^G}$ and study endomorphisms 
$\tau \colon \Sigma \to \Sigma $. We generalize several results for dynamical invariant sets and nilpotency of 
$\tau $ that are well known for finite alphabet cellular automata. Under mild assumptions, we prove that 
$\tau $ is nilpotent if and only if its limit set, that is, the intersection of the images of its iterates, is a singleton. If moreover G is infinite, finitely generated and 
$\Sigma $ is topologically mixing, we show that 
$\tau $ is nilpotent if and only if its limit set consists of periodic configurations and has a finite set of alphabet values.
We characterize Hermitian cones among the surfaces of degree 
$q+1$ of 
$\text{PG}(3,q^{2})$ by their intersection numbers with planes. We then use this result and provide a characterization of nonsingular Hermitian varieties of 
$\text{PG}(4,q^{2})$ among quasi-Hermitian ones.
