The problem of the initial stage of vertical and cavitational motion of a circular cylinder under the free surface of an ideal, incompressible, heavy fluid is considered. The dynamics of the attached cavity at short times is studied taking into account the boundary layer solutions at the separation points. The problem is considered to be correctly stated if the Kutta – Zhukovsky condition is satisfied at the separation points and the pressure in the liquid is a positive value. In the general case, the problem is considered taking into account artificial cavitation. In this case, an important role is played by the dynamics of the separation points, for which a special change of variables is made, fixing their position. The question of the possibility of representing the solution of the transformed problem as an asymptotic expansion in integer powers of short time is investigated. It is shown that the desired expansion containing the first two terms of the asymptotics exists only for certain physical situations. In the case of the simplest law of artificial cavitation, when the pressure in the cavity is a constant, these situations are characterized by a well-defined Froude number, which is equal to unity. In the general case, the power structure of the solution of the transformed problem at small times can be preserved if the corresponding boundary function is smoothed in the second asymptotic approximation. In this paper, much attention is paid to the behavior of the internal free boundary of the fluid (the boundary of the cavity) near the separation point. It is shown that in the leading approximation in time, this boundary approaches the separation point at a right angle (the square root of the difference of the corresponding angular coordinates arises). Refinement of this leading approximation for more moderate times leads to different patterns of fluid flow near the separation point. In some cases, the curve smoothing the square root comes out of the separation point and is located only on one side of it. In other cases, the curve near the separation point is located on different sides of it.