Volume 25, Issue 2

For any polyhedral norm, the bisector of two points is a polyhedral complex. We study combinatorial aspects of this complex. We investigate the sensitivity of the presence of labelled maximal cells in the bisector relative to the position of the two points. We thereby extend work of Criado, Joswig and Santos (2022) who showed that for the tropical distance function the presence of maximal cells is encoded by a polyhedral fan, the bisection fan. We initiate the study of bisection cones and bisection fans with respect to arbitrary polyhedral norms. In particular, we show that the bisection fan always exists for polyhedral norms in two dimensions. Furthermore, we determine the bisection fan of the ℓ 1 -norm and the ℓ ∞ -norm as well as the discrete Wasserstein distance in arbitrary dimensions. Intricate combinatorial structures, such as the resonance arrangement, make their appearance. We apply our results to obtain bounds on the combinatorial complexity of the bisectors.

In this note we give explicit constructions of decomposable hyperelliptic Jacobian varieties over fields of characteristic 0. These include hyperelliptic Jacobian varieties that are isogenous to a product of two absolutely simple hyperelliptic Jacobian varieties, a square of a hyperelliptic Jacobian variety, and a product of four hyperelliptic Jacobian varieties three of which are of the same dimension. As an application, we produce families of hyperelliptic curves with infinitely many quadratic twists having at least two rational non-Weierstrass points, and families of quadruples of hyperelliptic curves together with infinitely many non-square d such that the quadratic twists of each of the curves by d possess at least one rational non-Weierstrass point.

We obtain rigidity and triviality results for q -solitons, assuming that the structure tensor q satisfies a condition analogous to the twice contracted second Bianchi identity (in particular, divergence free and trace free) and that the soliton vector field is projective and an infinitesimal harmonic transformation. In particular, the efficacy of our general results is manifested by applying it to Bach solitons. Finally, we generalize the evolution equation for the scalar curvature of Ricci solitons to q -solitons in order to recover the Bourguignon–Ezin formula and a rigidity result on compact Ricci solitons.

Bowden, Hensel and Webb constructed infinitely many quasimorphisms on the diffeomorphism groups of orientable surfaces. In this paper, we extend their result to nonorientable surfaces. Namely, we prove that the space of nontrivial quasimorphisms QH˜Diff0Ng $\widetilde{Q H}\left(\operatorname{Diff}_0\left(N_g\right)\right)$on the identity component of the diffeomorphism group Diff 0 ( N g ) on a closed nonorientable surface N g of genus g ≥ 3 is infinite-dimensional. As a corollary, we obtain the unboundedness of the commutator length and the fragmentation length on Diff 0 ( N g ).

Let ε < 1/324 and let Ω be a smooth embedded paper Moebius band of aspect ratio less than 3+ε. $\sqrt{3}+\varepsilon .$We prove that Ω is within Hausdorff distance 18ε $18 \sqrt{\varepsilon}$of an equilateral triangle of perimeter 23. $2 \sqrt{3} .$This is an effective and fairly sharp version of our recent theorems in [6] about the optimal paper Moebius band.

Kahn [3] showed in 1980 that the Full Bundle Theorem characterizes ovoidal Laguerre planes. The spears and cycles (circles) of a general Laguerre plane can be represented by affine planes and points, respectively, of a near-linear space. In this space the Full Bundle Theorem takes a form analogous to the Veblen–Young Axiom for projective spaces. A proof that a form of Desargues’ Theorem within this near-linear space is equivalent to the Full Bundle Theorem is provided. Thus a Laguerre plane is desarguesian, in the sense of this paper, if, and only if, it is ovoidal.

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 C X of X is supported in the subcone 𝕍( Ω X [−1]) of its intrinsic normal sheaf N X . This leads to an alternative proof of cone reduction by cosections for C X . We also discuss vanishing of simple obstructions under the Buchweitz–Flenner semiregularity map for sheaves.

This note generalizes the celebrated Bogomolov–Gieseker inequality for smooth projective surfaces over an algebraically closed field of characteristic zero to projective surfaces in arbitrary characteristic with canonical singularities. We also generalize to this context some classical applications of the Bogomolov–Gieseker inequality.

This paper proves that the simultaneous lattice packing-covering constant of an octahedron is 7/6. In other words, 7/6 is the smallest positive number r such that for every octahedron O centered at the origin there is a lattice Λ such that O + Λ is a packing in 𝔼 3 and rO + Λ is a covering of 𝔼 3 .

Motivated by the recent surge of interest in the geometry of hybrid spaces, we prove an Abel–Jacobi theorem for a metrized complex of Riemann surfaces, generalizing both the classical Abel–Jacobi theorem and its tropical analogue.

Let C be a comb-like curve over ℂ and let E be a vector bundle of rank n on C . In this paper, we investigate the criteria for the semistability of the restriction of E onto the components of C when E is given to be semistable with respect to a polarization w . As an application, assuming that each irreducible component of C is general in its moduli space, we investigate the w -semistability of kernel bundles on such curves, extending the results (completely for rank two and partially for higher rank) known in the case of a reducible nodal curve with two smooth components, but here using different techniques.

In 1955 B. Segre showed that any oval in a projective plane over a finite field of odd order is a conic. We give another proof of this theorem.