For spatial high resolution determining a position of a singularized molecule, which is excitable with excitation light for emission of luminescence light, in n spatial dimensions in a sample, a preliminary local area including the singularized molecule is determined The excitation light is directed onto the sample with an intensity distribution, which has a zero point and intensity increasing regions adjoining the zero point on both sides in each of the n spatial dimensions. At first, the zero point is arranged at preliminary positions on known sides of the preliminary local area. Then, present positions of the zero point are successively shifted into the preliminary local area in each of the n spatial dimensions depending on photons of the luminescence light which is quasi-simultaneously separately registered for the present positions of the zero point in that the zero point is repeatedly shifted between the present positions of the zero point.