Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg最新文献

We study lines on smooth cubic surfaces over the field of p-adic numbers, from a theoretical and computational point of view. Segre showed that the possible counts of such lines are 0, 1, 2, 3, 5, 7, 9, 15 or 27. We show that each of these counts is achieved. Probabilistic aspects are investigated by sampling both p-adic and real cubic surfaces from different distributions and estimating the probability of each count.We link this to recent results on probabilistic enumerative geometry. Some experimental results on the Galois groups attached to p-adic cubic surfaces are also discussed.

We call a smooth irreducible projective curve a Castelnuovo curve if it admits a birational map into the projective r-space such that the image curve has degree at least 2r+1 and the maximum possible geometric genus (which one can calculate by a classical formula due to Castelnuovo). It is well known that a Castelnuovo curve must lie on a Hirzebruch surface (rational ruled surface). Conversely, making use of a result of W. Castryck and F. Cools concerning the scrollar invariants of curves on Hirzebruch surfaces we show that curves on Hirzebruch surfaces are Castelnuovo curves unless their genus becomes too small w.r.t. their gonality. We analyze the situation more closely, and we calculate the number of moduli of curves of fixed genus g and fixed gonality k lying on Hirzebruch surfaces, in terms of g and k.

In 2020 S. M. Gonek, S. W. Graham and Y. Lee formulated the Lindelöf hypothesis for prime numbers and proved that it is equivalent to the Riemann Hypothesis. In this note we show that their result holds in the Selberg class of L-functions.

We find and discuss an unexpected (to us) order n cyclic group of automorphisms of the Lie algebra (I{mathfrak u}_n{:}{=}{mathfrak u}_n < imes {mathfrak u}_n^*), where ({mathfrak u}_n) is the Lie algebra of upper triangular (ntimes n) matrices. Our results also extend to (mathfrak {gl}_{n+}^epsilon ), a “solvable approximation” of (mathfrak {gl}_n), as defined within.

We show that every compact complex analytic space endowed with a fine logarithmic structure and every morphism between such spaces admit a semi-universal deformation. These results generalize the analogous results in complex analytic geometry first independently proved by A. Douady and H. Grauert in the ’70. We follow Douady’s two steps process approach consisting of an infinite-dimensional construction of the deformation space followed by a finite-dimensional reduction.

In this paper, we firstly generalize the Brunn–Minkowski type inequality for Ekeland–Hofer–Zehnder symplectic capacity of bounded convex domains established by Artstein-Avidan–Ostrover in 2008 to extended symplectic capacities of bounded convex domains constructed by authors based on a class of Hamiltonian non-periodic boundary value problems recently. Then we introduce a class of non-periodic billiards in convex domains, and for them we prove some corresponding results to those for periodic billiards in convex domains obtained by Artstein-Avidan–Ostrover in 2012.

We study the nontrivial elements in the Brauer group of a bielliptic surface and show that they can be realized as Azumaya algebras with a simple structure at the generic point of the surface. We go on to study some properties of the noncommutative Picard scheme associated to such an Azumaya algebra.

We study the existence and uniqueness of renormalized solutions for initial boundary value problems of the type

where (u_{0}in L^{1}(Omega )), (mu in {mathcal {M}}_{b}(Q)) is a general Radon measure on Q and (Hin C_{b}^{0}({mathbb {R}})) is a continuous positive bounded function on ({mathbb {R}}). The difficulties in the study of such problems concern the possibly very singular right-hand side that forces the choice of a suitable formulation that ensures both existence and uniqueness of solution. Using similar techniques, we will prove existence/nonexistence results of the auxiliary problem

under the assumption that g satisfies a sign condition and the nonlinear term depends on both x, u and its gradient. Thus, our results improve and complete the previous known existence results for problems (left( {mathcal {P}}_{b}^{1,2}right) ).

Carmesin has extended Robertson and Seymour’s tree-of-tangles theorem to the infinite tangles of locally finite infinite graphs. We extend it further to the infinite tangles of all infinite graphs. Our result has a number of applications for the topology of infinite graphs, such as their end spaces and their compactifications.

In this article we study the relation between flat solvmanifolds and (G_2)-geometry. First, we give a classification of 7-dimensional flat splittable solvmanifolds using the classification of finite subgroups of (mathsf{GL}(n,mathbb {Z})) for (n=5) and (n=6). Then, we look for closed, coclosed and divergence-free (G_2)-structures compatible with the flat metric on them. In particular, we provide explicit examples of compact flat manifolds with a torsion-free (G_2)-structure whose finite holonomy is cyclic and contained in (G_2), and examples of compact flat manifolds admitting a divergence-free (G_2)-structure.