Geometriae Dedicata

In this work, we study the cellular decomposition of S induced by a filling pair of curves v and w, $$Dec_{v,w}(S) = S {\setminus } (v \cup w)$$ , and its connection to the distance function d(v, w) in the curve graph of a closed orientable surface S of genus g. Building on the work of Leasure, efficient geodesics were introduced by the first author in joint work with Margalit and Menasco in 2016, giving an algorithm that begins with a pair of non-separating filling curves that determine vertices (v, w) in the curve graph of a closed orientable surface S and computing from them a finite set of efficient geodesics. We extend the tools of efficient geodesics to study the relationship between distance d(v, w), intersection number i(v, w), and $$Dec_{v,w}(S)$$ . The main result is the development and analysis of particular configurations of rectangles in $$Dec_{v,w}(S)$$ called spirals. We are able to show that, with appropriate restrictions, the efficient geodesic algorithm can be used to build an algorithm that reduces i(v, w) while preserving d(v, w). At the end of the paper, we note a connection between our work and the notion of extending geodesics.

Classical H. Minkowski theorems on existence and uniqueness of convex polyhedra with prescribed directions and areas of faces as well as the well-known generalization of H. Minkowski uniqueness theorem due to A.D. Alexandrov are extended to a class of nonconvex polyhedra which are called polyhedral herissons and may be described as polyhedra with injective spherical image.

We investigate the geometry of word metrics on fundamental groups of manifolds associated with the generating sets consisting of elements represented by closed geodesics. We ask whether the diameter of such a metric is finite or infinite. The first answer we interpret as an abundance of closed geodesics, while the second one as their scarcity. We discuss examples for both cases.

Let (X,L) be a quasi-polarized variety, i.e. X is a smooth projective variety over the complex numbers $$\mathbb{C}$$ and L is a nef and big divisor on X. Then we conjecture that g(L) = q(X), whereg(L) is the sectional genus of L and $$q(X) = \dim H^1 (\mathcal{O}_X )$$ . In this paper, we treat the case $$\dim X = 2$$ . First we prove that this conjecture is true for $$\kappa (X) \leqslant 1$$ , and we classify (X,L) withg(L)=q(X), where $$\kappa (X)$$ is the Kodaira dimension of X. Next we study some special cases of $$\kappa (X) = 2$$ .

In this paper we characterize smooth complex projective varieties that admit a quadric bundle structure on some dense open subset in terms of the geometry of certain families of rational curves.

We prove the infinitesimal rigidity of some geometrically infinite hyperbolic 4- and 5-manifolds. These examples arise as infinite cyclic coverings of finite-volume hyperbolic manifolds obtained by colouring right-angled polytopes, already described in the papers (Battista in Trans Am Math Soc 375(04):2597–2625, 2022; Italiano et al. in Invent Math, 2022. https://doi.org/10.1007/s00222-022-01141-w ). The 5-dimensional example is diffeomorphic to $$N\times {{\mathbb {R}}}$$ for some aspherical 4-manifold N which does not admit any hyperbolic structure. To this purpose, we develop a general strategy to study the infinitesimal rigidity of cyclic coverings of manifolds obtained by colouring right-angled polytopes.