The revealing of the "diodelike" properties of electrolyte-filled asymmetric nanopores in track membranes has given significant impetus to a detailed study of the properties of "track" nanocapillaries. Studying the behavior of electrolyte solutions in nanovolumes of a given geometry is very important for many applications, such as nanofluid technology, the resistive pulse method for detecting colloidal particles and molecules, modeling of biological membranes, etc. An attempt to find a quantitative relationship between the geometric shape of asymmetric nanopores and asymmetry in electrical conductivity has been made in this paper. The method of chemical etching in the presence of a surfactant was used for the formation of nanopores with different profiles. The pore structure was studied by electron microscopy. It has been found that the rectification ratio increases with the membrane thickness and depends strongly on the curvature of the pore profile in the selective layer. The maximum of the rectification has been observed in a 0.05-0.1M KCl. Simulation of the ionic conductivity of asymmetric nanopores by the Poisson-Nernst-Planck equation qualitatively explains the observed behavior. The effect of the asymmetry of electrical conductivity is well expressed even in cases when the pore radius in the selective layer is substantially greater than the Debye length. The modification of the pore surface by grafting of aminopropyltriethoxysilane results in the sign inversion of electric charge and a sharp change in the current-voltage characteristics of the membranes.