Embedding structure of projective varieties and the initial ideal of their definig equations

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The Castelnuovo-Mumford regularity of a projective varierty X refrects its defining equations, generic initial ideal of defining ideal, and the Hilbert function of X. On the other hand, the regularity is expected to have a strong relation to the existence of multisecant lines to X. From this point of view, in this period, for an irreducible, projective variety X of degree d and codimension e, defined over an algebraically closed field, we study (I) multisecant lines to X; (II) hypersurfaces of small degree containing X. In (II), in particular, letting E(X) be the intersection of all hypersurfaces of degree at most d-e+1, containing X, we study if X = E(X) as an evidence of the regularity conjecture. Let B(X) be the points of outside of X, from which the projection of X is not birational onto its image. Similarly, let C(X) be the smooth points of X, from which the projection of X is not birational onto its image. We have the following results.(I-1) If X is smooth of sectional genus g, the length of the intersection of X and a line does not exceed d-e+1-g.(I-2) The length of the intersection of X and a line does not exceed d-e+1 if the projection of X from the line is quasi-finite.(II-1) As sets, X=E(X) outside of B(X), and as schemes, X=E(X) outside of B(X), C(X) and the singular locus Sing(X) of X.(II-2) The dimension of B(X) does not exceed the dimension of Sing(X) puls 1. Moreover, the dimension of C(X) does not exceed the dimension of Sing(X) plus 2.