algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety hypersurfaces; Schebert calculus; cubic hypersurfaces; cubic threefolds; cubic fourfolds; Pfaffian cubics; unirationality; rationality; Picard group; intermediate Jacobian; Albanese variety; Abel-Jacobi map; conic bundles; abelian varieties; Prym varieties; Hilbert square; varieties with vanishing Chern class; Calabi-Yau varieties; holomorphic symplectic varieties; Beauville-Bogomolov decomposition theorem 