Unirationality of Hurwitz spaces of coverings of degree <= 5

Let $Y$ be a smooth, projective curve of genus $g\geq 1$ over the complex numbers. Let $H^0_{d,A}(Y)$ be the Hurwitz space which parametrizes coverings $p:X \to Y$ of degree $d$, simply branched in $n=2e$ points, with monodromy group equal to $S_d$, and $det(p_{*}O_X/O_Y)$ isomorphic to a fixed line bundle $A^{-1}$ of degree $-e$. We prove that, when $d=3, 4$ or $5$ and $n$ is sufficiently large (precise bounds are given), these Hurwitz spaces are unirational. If in addition $(e,2)=1$ (when $d=3$), $(e,6)=1$ (when $d=4$) and $(e,10)=1$ (when $d=5$), then these Hurwitz spaces are rational.Comment: Proposition 2.11 and Lemma 2.13 are corrected. The corrections do not affect the other statements of the paper. Corrigendum submitted to IMR

The q−q-state Potts model from the Nonperturbative Renormalization Group

We study the $q$-state Potts model for $q$ and the space dimension $d$ arbitrary real numbers using the Derivative Expansion of the Nonperturbative Renormalization Group at its leading order, the local potential approximation (LPA and LPA'). We determine the curve $q_c(d)$ separating the first ($q>q_c(d)$) and second ($q<q_c(d)$) order phase transition regions for $2.8<d\leq 4$. At small $\epsilon=4-d$ and $\delta=q-2$ the calculation is performed in a double expansion in these parameters and we find $q_c(d)=2+a \epsilon^2$ with $a\simeq 0.1$. For finite values of $\epsilon$ and $\delta$, we obtain this curve by integrating the LPA and LPA' flow equations. We find that $q_c(d=3)=2.11(7)$ which confirms that the transition is of first order in $d=3$ for the three-state Potts model.Comment: 21 pages, 10 figures. The source file includes Supplementary Materia

Neutral low-dimensional assemblies of a Mn(III) schiff base complex and octacyanotungstate(V) : synthesis, characterization, and magnetic properties

International audienceTwo novel low-dimensional molecular magnetic materials were prepared by the self-assembly of 3d- and 5d-metal complexes. These are the first neutral heterobimetallic cyanobridged compounds involving one anisotropic Mn(III) Schiff base complex and one octacyanotungstate(V) per molecular unit. A slow diffusion of the constituents’ solutions leads to the formation of the 0D crystalline complex 1, due to coordination of a water molecule to the Mn center, which prevents polymer formation. A rapid mixing of reagents results in the precipitation of the microcrystalline powder of complex 2, which based on the totality of experimental data, possesses a 1D polymeric structure. The magnetic studies have shown that antiferromagnetic exchange interactions prevail in 1 (J/kB = −13.1(7) K, D = −3.0(1.3) K, zJ' = −0.16(20) K and gav = 2.00(1)); while the presence of the significant intramolecular Mn(III)–W(V) ferromagnetic coupling through cyanide bridge is characteristic for 2 (J/kB = 46.1(5) K, gMn = 2.11(3), fixed gW = 2.0). Due to the weak interchain interactions, zJ′/kB = −0.8(2) K, and compound 2 is a metamagnet with the Néel temperature of 9.5 K undergoing a spin-flip transition at 2 kOe. The slow magnetization dynamics of 2 were investigated at a DC field of 0 and 2 kOe, giving the values of τ0 32(15) and 36(15) ps, respectively, well within the range typical for single-chain magnets (SCMs). The respective ∆τ/kB values were 48.4(1.2) and 44.9(1.0) K

Real polynomials with constrained real divisors. I. Fundamental groups

In the late 80s, V.~Arnold and V.~Vassiliev initiated the topological study of the space of real univariate polynomials of a given degree d and with no real roots of multiplicity exceeding a given positive integer. Expanding their studies, we consider the spaces of real monic univariate polynomials of degree d whose real divisors avoid sequences of root multiplicities taken from a given poset of compositions which is closed under certain natural combinatorial operations. In this paper, we concentrate on the fundamental group of such spaces. We find explicit presentations for the fundamental groups in terms of generators and relations and show that in a number of cases they are free with rank bounded from above by a quadratic function in d. We also show that the fundamental group stabilizes for d large. We further show that the fundamental groups admit an interpretation as special bordisms of immersions of 1-manifolds into the cylinder S^1 \times R, whose images avoid the tangency patterns from the poset with respect to the generators of the cylinder.Comment: Section 2.3 completely rewritten. Assumptions of Theorem 2.11 corrected and new results on stabilization added. Additional minor change