Abstract LetX⊂ ℙnbe an unramified real curve withX(ℝ) ≠ 0. Ifn≥ 3 is odd, Huisman [9] conjectured thatXis anM-curve and that every branch ofX(ℝ) is a pseudo-line. Ifn≥ 4 is even, he conjectures thatXis a rational normal curve or a twisted form of such a curve. Recently, a family of unramifiedM-curves in ℙ3providing counterexamples to the first conjecture was constructed in [11]. In this note we construct another family of counterexamples that are not evenM-curves. We remark that the second conjecture follows for generic curves of odd degree from the de Jonquières formula.

Abstract We study the automorphisms of a function field of genus g ≥ 2 over an algebraically closed field of characteristic p > 0. More precisely, we show that the order of a nilpotent subgroup G of its automorphism group is bounded by 16 (g – 1) when G is not a p-group. We show that if |G| = 16(g – 1), then g – 1 is a power of 2. Furthermore, we provide an infinite family of function fields attaining the bound.

Abstract Let X be an Enriques surface over the field of complex numbers. We prove that there exists a nontrivial quaternion algebra 𝓐 on X. Then we study the moduli scheme of torsion free 𝓐-modules of rank one. Finally we prove that this moduli scheme is an étale double cover of a Lagrangian subscheme in the corresponding moduli scheme on the associated covering K3 surface.

Abstract We present various results about the combinatorial properties of line arrangements in terms of the Chern numbers of the corresponding log surfaces. This resembles the study of the geography of surfaces of general type. We prove some new results about the distribution of Chern slopes, we show a connection between their accumulation points and the accumulation points of linear H-constants on the plane, and we conclude with two open problems in relation to geography over ℚ and over ℂ.

Abstract LetFbe the finite field of orderq2. In this paper we continue the study in [24], [23], [22] ofF-maximal curves defined by equations of typeyn=xℓ(xm+1).${{y}^{n}}={{x}^{\ell }}\left( {{x}^{m}}+1 \right).$New results are obtained via certain subcovers of the nonsingular model ofvN=ut2−u${{v}^{N}}={{u}^{{{t}^{2}}}}-u$whereq=tα,α≥ 3 is odd andN= (tα+ 1)/(t+ 1). We observe that the caseα= 3 is closely related to the Giulietti–Korchmáros curve.

Abstract In the Euclidean plane, two circles that intersect or are tangent clearly do not carry a finite Steiner chain of circles. We show that such exotic Steiner chains exist in finite Miquelian Möbius planes of odd order. We obtain explicit conditions in terms of the order of the plane and the capacitance of the two carrier circles for the existence, length, and number of Steiner chains.

Abstract We classify 3-dimensional compact Riemannian manifolds (M 3, g) that admit a non-constant solution to the equation −Δfg +Hess f − f Ric = μ Ric +λg for some special constants (μ, λ), under the assumption that the manifold has cyclic parallel Ricci tensor. Namely, the structures that we study here are: positive static triples, critical metrics of the volume functional, and critical metrics of the total scalar curvature functional. We also classify n-dimensional critical metrics of the volume functional with non-positive scalar curvature and satisfying the cyclic parallel Ricci tensor condition.

Abstract Let S be a surface of genus g at least 2. A representation ρ : π 1 S → P S L 2 R $ \rho:\pi_1 S\to{\mathrm{PSL}_2\mathbb{R}} $ is said to be purely hyperbolic if its image consists only of hyperbolic elements along with the identity. We may wonder under which conditions such representations arise as the holonomy of a branched hyperbolic structure on S. In this work we characterise them completely, giving necessary and sufficient conditions.

Abstract Some coherent sheaves on projective varieties have a non-reduced versal deformation space; for example, this is the case for most unstable rank 2 vector bundles on ℙ2, see [18]. In particular, some moduli spaces of stable sheaves are non-reduced. We consider some sheaves on ribbons (double structures on smooth projective curves): let E be a quasi locally free sheaf of rigid type and let 𝓔 be a flat family of sheaves containing E. We find that 𝓔 is a reduced deformation of E when some canonical family associated to 𝓔 is also flat. We consider also a deformation of the ribbon to reduced projective curves with two components, and find that E can be deformed in two distinct ways to sheaves on the reduced curves. In particular some components M of the moduli spaces of stable sheaves deform to two components of the moduli spaces of sheaves on the reduced curves, and M appears as the “limit” of varieties with two components, whence the non-reduced structure of M.

Abstract Let 𝔽 = ℝ, ℂ or ℍ. A p-set of equi-isoclinic n-planes with parameter λ in 𝔽 r is a set of p n-planes spanning 𝔽 r each pair of which has the same non-zero angle arccos λ $\begin{array}{} \sqrt{\lambda} \end{array}$ . It is known that via a complex matrix representation, a pair of isoclinic n-planes in ℍ r with angle arccos λ $\begin{array}{} \sqrt{\lambda} \end{array}$ yields a pair of isoclinic 2n-planes in ℂ2r with angle arccos λ $\begin{array}{} \sqrt{\lambda} \end{array}$ . In this article we characterize all the p-tuples of equi-isoclinic planes in ℂ2r which come via our complex representation from p-tuples of equiangular lines in ℍ r . We then construct all the p-tuples of equi-isoclinic planes in ℂ4 and derive all the p-tuples of equiangular lines in ℍ2. Among other things it turns out that the quadruples of equiangular lines in ℍ2 are all regular, i.e. their symmetry groups are isomorphic to the symmetric group S 4.