Abstract The classical Brill–Noether theorem states that a map from a general curve to a projective space deforms in a family of the expected dimension if its image does not lie in any hyperplane. In this note, we observe, as a direct consequence of standard results on Severi varieties, an analogous statement for maps from a general curve to any smooth projective surface. Namely, a non-constant map deforms in a family of the expected dimension if its image has anti-canonical degree at least 4. In the case of toric surfaces, curves of anti-canonical degree at most 3 admit a particularly elegant description in terms of certain toric contractions. We raise the question of whether a Brill–Noether theorem could hold for toric varieties of higher dimensions.

Abstract We revisit the theory of Kac–Moody symmetric spaces (of "non-compact type") in order to include the affine case, complex numbers, and Galois descent to almost split real Kac–Moody symmetric spaces.

Abstract In this paper we prove that there is a flock in every locally compact 2-dimensional Laguerre plane, i.e., a set of disjoint circles which covers the point space. We then determine the homeomorphism type of these flocks and use the relation between circle planes and generalised quadrangles to obtain analogous results about generalised quadrangles, Möbius planes and Minkowski planes.

Abstract The inscribed square problem asks if every Jordan curve contains the four vertices of a non-degenerate Euclidean square. This question was posed by Otto Toeplitz in 1911 and remains open to this date. It is known that, with the exception of the arc and the 3-star, every topological copy of a locally connected plane continuum in ℝ 2 contains the four vertices of at least one Euclidean rectangle; see [13]. Given ε > 0, a hyperbolic ε -rectangle is a hyperbolic quadrilateral such that its inner angles sum up to more than 2 π − ε , and such that its diagonals share the midpoint and have the same hyperbolic length. A tree T quasi-inscribes rectangles if for every ε > 0, any embedding of T in the hyperbolic plane admits an inscribed ε -rectangle. In this paper, we classify trees that quasi-inscribe rectangles. We also show that any Jordan curve J in the hyperbolic plane admits at least one equilateral hyperbolic triangle whose area is less than π − K , where K is a positive constant that depends only on the diameter of J .

Abstract This paper develops the geometry of locally bounded rational functions on non-singular real algebraic varieties. First various basic geometric and algebraic results regarding these functions are established in any dimension, culminating with a version of Łojasiewicz’s inequality. The geometry is further developed for the case of dimension 2, where it can be shown that there exist many of the usual correspondences between the algebra and geometry of these functions that one expects from complex algebraic geometry and from other classes of functions in real algebraic geometry such as regulous functions.

Abstract Let X be a Deligne–Mumford stack locally of finite type over an algebraically closed field k of characteristic zero. We show that the intrinsic normal cone CX of X is supported in the subcone 𝕍(ΩX [−1]) of its intrinsic normal sheaf NX . This leads to an alternative proof of cone reduction by cosections for CX . We also discuss vanishing of simple obstructions under the Buchweitz–Flenner semiregularity map for sheaves.

Abstract A p-set of equiangular lines in ℂ3 is a set of p lines spanning ℂ3 each pair of which has the same nonzero angle arccos c, where 0 < c < 1. It is known that via a real matrix representation, a pair of lines in ℝ3 with angle arccos c yields a pair of isoclinic planes in ℝ6 with angle arccos c. In this article we characterize all p-tuples of equi-isoclinic planes in ℝ6 which come via our real matrix representation from p-tuples of equiangular lines in ℂ3. More precisely, we describe quadruples, quintuples up to nine-tuples of equiangular lines in ℂ3.

Abstract Let n ≥ 3 and let K ⊂ ℝ n be a convex body. A point p ∈ int K is said to be a Larman point of K if for every hyperplane Π passing through p, the section Π ∩ K has an (n – 2)-plane of symmetry. If p is a Larman point of K and for every section Π ∩ K, p is in the corresponding (n – 2)-plane of symmetry, then we call p a revolution point of K. We conjecture that if K contains a Larman point which is not a revolution point, then K is either an ellipsoid or a body of revolution. This generalizes a conjecture of Bezdek for n = 3. We prove several results related to the conjecture for strictly convex origin symmetric bodies. Namely, if K ⊂ ℝ n is a strictly convex origin symmetric body that contains a revolution point p which is not the origin, then K is a body of revolution. This generalizes the False Axis of Revolution Theorem in [7]. We also show that if p is a Larman point of K ⊂ ℝ3 and there exists a line L such that p ∉ L and, for every plane Π passing through p, the line of symmetry of the section Π ∩ K intersects L, then K is a body of revolution (in some cases, K is a sphere). We obtain a similar result for projections of K. Additionally, for K ⊂ ℝ n with n ≥ 4, we show that if every hyperplane section or projection of K is a body of revolution and K has a unique diameter D, then K is a body of revolution with axis D.

Abstract On a 7-manifold with a G 2-structure, we study conformal symmetries — which are vector fields whose flow generate conformal transformations of the G 2-structure. In particular, we focus on compact 7-manifolds and the condition that the Lee form of the G 2-structure is closed. Among other observations, we show that conformal symmetries are determined within a conformal class of the G 2-structure by the symmetries of a unique (up to homothety) G 2-structure whose Lee form is harmonic. On a related note, we also demonstrate that symmetries are split along fibrations when the Lee vector field is itself a symmetry.