We show that for any smooth Calabi–Yau threefold $X$ of Picard number $2$ with infinite birational automorphism group, the numerical dimension $\kappa _\sigma $ of the extremal rays of the movable cone of $X$ is $\frac {3}{2}$. Furthermore, we provide new examples of Calabi–Yau threefolds of Picard number $2$ with infinite birational automorphism group.

We study Calabi-Yau manifolds which are complete intersections of hypersurfaces of multidegree 1 in an m-fold product of n-dimensional projective spaces. Using the theory of Coxeter groups, we show that the birational automorphism group of such a Calabi-Yau manifold X is infinite and a free product of copies of Z . Moreover, we give an explicit description of the boundary of the movable cone Mov(X). In the end, we consider examples for the general and non-general case and picture the movable cone and the fundamental domain for the action of Bir(X).

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The famous structure theorem of Buchsbaum and Eisenbud gives a complete characterization of Gorenstein ideals of codimension 3 and their minimal free resolutions. We generalize the ideas of Buchsbaum and Eisenbud from Gorenstein ideals to Gorenstein algebras and present a description of Gorenstein algebras of any odd codimension. As an application we study the canonical ring of a numerical Godeaux surface.