The separability of the Gauss map and the reflexivity for a projective surface

Abstract

It is known that if a projective variety X in P N is reflexive with respect to the projective dual, then the Gauss map of X defined by embedded tangent spaces is separable, and moreover that the converse is not true in general. We prove that the converse holds under the assumption that X is of dimension two. Explaining the subtleness of the problem, we present an example of smooth projective surfaces in arbitrary positive characteristic, which gives a negative answer to a question raised by S. Kleiman and R. Piene on the inseparability of the Gauss map.

The separability of the Gauss map versus the reflexivity

“…However, the situation in the two-dimensional case seems to be not so simple and more subtle: In fact, we have an example of a projective surface which gives a negative answer to Question 2.4: 26]) Let X be a smooth projective surface in P 10 given by the image of an embedding ι : P 2 → P 10 defined by ι(1 : x : y) := (1 : x : y : x p : y p : x p 2 : y p 2 : x p 2 +1 : x y p 2 : x p 2 y : y p 2 +1 ).…”

The reflexivity of a Segre product of projective varieties

“…Our Theorem 1.1 yields examples answering Kleiman-Piene's question in the negative, as mentioned in the Introduction: For instance, the first known example given by the second author [7] falls within Theorem 1.1 (1), and the example given by the first author [1] falls within (2). Note that every non-reflexive P m × Y in Table 1 is a negative example to Kleiman-Piene's question by Proposition 3.5.…”

“…Then it is not difficult to see that a general Hessian matrix of Y has rank r = 0 and Y is non-reflexive (see, for example, the formulae (1), (2) and (3) in [2]). A Fermat hypersurface F d is in the case if d ≡ 1 mod p. Some other examples are found in [3]: In fact, consider the graph F of a Frobenius morphism F of a projective space P n for instance, and embed F into P N with N + 1 = (n + 1) 2 by the Segre embedding.…”

“…Here, the Gauss map γ of X is defined to be a rational map from X to a Grassmann variety G(n, P N ) which assigns to each smooth point P ∈ X the projective tangent space T P X , where n is the dimension of X . It is easily seen that the conormal map π factors as a composition of a base change of γ and a projection, denoted by π , from a certain incidence correspondence over G(N − n − 1,P N ) ∼ = G(n, P N ) onto the dual X * [1,2,6,9]. Then, the question of Kleiman and Piene [9, pp.…”

Existence of a non‐reflexive embedding with birational Gauss map for a projective variety

“…For a projective curve X, it is well-known that the generic smoothness of the Gauss map γ is equivalent to that of the conormal map π ([5], [6], [8], [9], [14]). It is also known that the generic smoothness of π implies that of γ for any projective variety X ( [12]), and that the converse does not hold in general if and only if dim X ≥ 3 ([2], [3], [4], [10]). Thus the relationship between the generic smoothness of the Gauss map γ and of the conormal map π for a fixed projective embedding of X has been studied well.…”