A NOTE ON TODOROV SURFACES

Let S be a Todorov surface, i.e., a minimal smooth surface of general type with q = 0 and p(g) = 1 having an involution i such that S/i is birational to a K3 surface and such that the bicanonical map of S is composed with i. The main result of this paper is that, if P is the minimal smooth model of S/i, then P is the minimal desingularization of a double cover of P(2) ramified over two cubics. Furthermore it is also shown that, given a Todorov surface S, it is possible to construct Todorov surfaces S(j) with K(2) = 1,..., K(S)(2) - 1 and such that P is also the smooth minimal model of S(j)/i(j), where i(j) is the involution of S(j). Some examples are also given, namely an example different from the examples presented by Todorov in [9].