We give a solution to the weak Schottky problem for genus five Jacobians with vanishing theta null, answering question of Grushevsky and Salvati Manni. More precisely, we show that if principally polarized abelian variety dimension has null quadric tangent cone rank at most three, then it is in Jacobian locus, up extra irreducible components. employ degeneration argument, together study ramific...

Fix a prime power q and integers m ≥ 2, σ > 0, x > 0, g ≥ 0, d ≥ mσ + 1 such that q ≥ (δ − 1)δ3, where δ := d3 + 3dx + 4x + 2g − 2. Let C ⊂ P be a smooth degree x curve defined over Fq such that h(P,IC(σ − 1)) = h(C,OC (σ − 2)) = 0 and pa(C) = g. Here we prove the existence of a degree d surface X ⊆ P3 defined over Fq, such that Sing(X) = C and X has ordinary multiplicity m along C, i.e. for ev...

We investigate the relationships between the Lipschitz outer geometry and the embedded topological type of a hypersurface germ in (Cn, 0). It is well known that the Lipschitz outer geometry of a complex plane curve germ determines and is determined by its embedded topological type. We prove that this does not remain true in higher dimensions. Namely, we give two normal hypersurface germs (X1, 0...

The geometry of circles in the plane is inextricably tied with the group of Möbius transformations, which take circles to circles. This geometry can be seen in a more symmetric after transforming the plane to the sphere, by stereographic projection. Interpretations will be discussed for Voronoi diagrams, Delaunay triangulations,etc. from this point of view. Fortune’s algorithm for constructing ...

The aim of this paper is to derive convergence results for projected line-search methods on the real-algebraic variety M≤k of real m× n matrices of rank at most k. Such methods extend Riemannian optimization methods, which are successfully used on the smooth manifold Mk of rank-k matrices, to its closure by taking steps along gradient-related directions in the tangent cone, and afterwards proje...

As an intial step towards low-rank optimization algorithms using hierarchical tensors, the aim of this paper is to derive convergence results for projected line-search methods on the real-algebraic variety M≤k of real m × n matrices of rank at most k. Such methods extend successfully used Riemannian optimization methods on the smooth manifold Mk of rank-k matrices to its closure by taking steps...

Let η be a line bundle on a smooth curve X with η^2=0 such that π_η, the double covering induced by η is an etale morphism. Assume also that X_η be the Prym-canonical model of X associated to K_X.η and Q is a rank 4 quadric containing X_η. After stablishing the projective normality of the prym-canonical models of curves X with Clifford index 2, we obtain in this paper a sufficient condition for...

In this letter, our objective is to explore how two well-known projection dynamics can be used as dynamic controllers for stabilization of nonlinear systems. Combining the properties operators, Lyapunov stability theory and LaSalle’s theorem, we confirm that on feasible set tangent cone are Krasovskii passive. To show effectiveness proposed approach, use stabilizing boost converters in a DC mic...