Osculation for conic fibrations.

Smooth projective surfaces fibered in conics over a smooth curve are investigated with respect to their k-th osculatory behavior. Due to the bound for the dimension of their osculating spaces they do not differ at all from a general surface for k = 2, while their structure plays a significant role for k >= 3. The dimension of the osculating space at any point is studied taking into account the possible existence of curves of low degree transverse to the fibers, and several examples are discussed to illustrate concretely the various situations arising in this analysis. As an application, a complete description of the osculatory behavior of Castelnuovo surfaces is given. The case k = 3 for del Pezzo surfaces is also discussed, completing the analysis done for k = 2 in a previous paper by the authors (2001). Moreover, for conic fibrations X subset of P-N whose k-th inflectional locus has the expected codimension, a precise description of this locus is provided in terms of Chern classes. In particular, for N = 8, it turns out that either X is hypo-osculating for k = 3, or its third inflectional locus is 1-dimensional

Generalized isotropic Lipkin-Meshkov-Glick models: ground state entanglement and quantum entropies

We introduce a new class of generalized isotropic Lipkin–Meshkov–Glick models with su(m+1) spin and long-range non-constant interactions, whose non-degenerate ground state is a Dicke state of su(m+1) type. We evaluate in closed form the reduced density matrix of a block of Lspins when the whole system is in its ground state, and study the corresponding von Neumann and Rényi entanglement entropies in the thermodynamic limit. We show that both of these entropies scale as a log L when L tends to infinity, where the coefficient a is equal to (m  −  k)/2 in the ground state phase with k vanishing magnon densities. In particular, our results show that none of these generalized Lipkin–Meshkov–Glick models are critical, since when L-->∞ their Rényi entropy R_q becomes independent of the parameter q. We have also computed the Tsallis entanglement entropy of the ground state of these generalized su(m+1) Lipkin–Meshkov–Glick models, finding that it can be made extensive by an appropriate choice of its parameter only when m-k≥3. Finally, in the su(3) case we construct in detail the phase diagram of the ground state in parameter space, showing that it is determined in a simple way by the weights of the fundamental representation of su(3). This is also true in the su(m+1) case; for instance, we prove that the region for which all the magnon densities are non-vanishing is an (m  +  1)-simplex in R^m whose vertices are the weights of the fundamental representation of su(m+1).