The motivic Igusa zeta function of a space monomial curve with a plane semigroup

In this article, we compute the motivic Igusa zeta function of a space monomial curve that appears as the special fiber of an equisingular family whose generic fiber is a complex plane branch. To this end, we determine the irreducible components of the jet schemes of such a space monomial curve. This approach does not only yield a closed formula for the motivic zeta function, but also allows to determine its poles. We show that, while the family of the jet schemes of the fibers is not flat, the number of poles of the motivic zeta function associated with the space monomial curve is equal to the number of poles of the motivic zeta function associated with a generic curve in the family.

Integrated robot planning, path following, and obstacle avoidance in two and three dimensions: Wheeled robots, underwater vehicles, and multicopters

We propose an innovative, integrated solution to path planning, path following, and obstacle avoidance that is suitable both for 2D and 3D navigation. The proposed method takes as input a generic curve connecting a start and a goal position, and is able to find a corresponding path from start to goal in a maze-like environment even in the absence of global information, it guarantees convergence to the path with kinematic control, and finally avoids locally sensed obstacles without becoming trapped in deadlocks. This is achieved by computing a closed-form expression in which the control variables are a continuous function of the input curve, the robot’s state, and the distance of all the locally sensed obstacles. Specifically, we introduce a novel formalism for describing the path in two and three dimensions, as well as a computationally efficient method for path deformation (based only on local sensor readings) that is able to find a path to the goal even when such path cannot be produced through continuous deformations of the original. The article provides formal proofs of all the properties above, as well as simulated results in a simulated environment with a wheeled robot, an underwater vehicle, and a multicopter.

On monodromy representation of period integrals associated to an algebraic curve with bi-degree (2,2)

We study a problem related to Kontsevich's homological mirror symmetry conjecture for the case of a generic curve Y with bi-degree (2,2) in a product of projective lines ℙ1× ℙ1. We calculate two differenent monodromy representations of period integrals for the affine variety X(2,2)obtained by the dual polyhedron mirror variety construction from Y. The first method that gives a full representation of the fundamental group of the complement to singular loci relies on the generalised Picard-Lefschetz theorem. The second method uses the analytic continuation of the Mellin-Barnes integrals that gives us a proper subgroup of the monodromy group. It turns out both representations admit a Hermitian quadratic invariant form that is given by a Gram matrix of a split generator of the derived category of coherent sheaves on on Y with respect to the Euler form.

Logarithmic Vector Fields and Hyperbolicity

Using vector fields on logarithmic jet spaces we obtain some new positive results for the logarithmic Kobayashi conjecture about the hyperbolicity of complements of curves in the complex projective plane. We are interested here in the cases where logarithmic irregularity is strictly smaller than the dimension. In this setting, we study the case of a very generic curve with two components of degrees d1 ≤ d2 and prove the hyperbolicity of the complement if the degrees satisfy either d1 ≥ 4, or d1 = 3 and d2 ≥ 5, or d1 = 2 and d2 ≥ 8, or d1 = 1 and d2 ≥ 11. We also prove that the complement of a very generic curve of degree d at least equal to 14 in the complex projective plane is hyperbolic, improving slightly, with a new proof, the former bound obtained by El Goul.

A Linear Haptic Interface for the Evaluation of Shapes

The paper presents a haptic device that allows a user to explore a virtual object along a continuous line. In particular the device is developed with the aim of supporting designers during the evaluation of the aesthetic quality of a virtual product. This is generally done by means of the global and local analysis of the shape in terms of curvature characteristics, presence of inflections points and discontinuities. In order to evaluate such features, designers are used to work on physical prototypes, relying on their skilled sense of touch. It is known that physical prototypes are expensive in terms of cost and time for their realization, and a modification on a physical prototype implies a reverse engineering process for appling such modifications on the virtual model. A linear haptic interface, that adapts its shape reproducing a generic curve on a surface, has been developed to replicate the behavior of a physical strip. This is the way to replace real prototypes with virtual ones without changing the evaluation paradigms that designers are used to. The physical limitations encountered in representing discontinuities in position, tangency and curvature, not renderable by bending and deforming a physical strip, have been overcome thanks to the application of some principles of the theory of haptic illusions by means of sonification of some curve characteristics. The paper describes the linear haptic interface developed and the solution based on haptic illusion that has been implemented to overcome the strip limitations.

Petri map for rank two bundles with canonical determinant

We prove the Bertram–Feinberg–Mukai conjecture for a generic curve C of genus g and a semistable vector bundle E of rank two and determinant K on C, namely we prove the injectivity of the Petri-canonical map S2(H0(E))→H0(S2(E)).