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Geoletry and algebra of lultldllenslonal three-webs I by Maks A. Aklvls and Alexander M. Shelekhov. p. c •. --(Mathelatlcs and lts appllcatlons (Sovlet serles) v. 82)Translated frol the Russlan.Includes blbllographlcal references and Index.

Let V = C N +1 and let X n ⊂ PV be a variety. Let x ∈ X be a smooth point, and let Tx X ⊂ PV denote the embedded tangent projective space to X at x. Let γ : X G(n, PV )x → Tx X denote the Gauss map of X, where G(n, PV ) denotes the Grasmannian of P n 's in PV .In [GH], Griffiths and Harris present a structure theorem for varieties with degenerate Gauss mappings, that is X such that dim γ(X) < dim X. Namely, such varieties are "built up from cones and developable varieties" [GH, p. 392]. By "built up from" they appear to mean "foliated by" and by "developable varieties" they appear to mean the osculating varieties to a curve. With these interpretations, their result appears to be complete for varieties whose Gauss maps have one-dimensional fibers. For varieties with higher dimensional fibers, one could generalize "built up from" to mean either foliated by, or iteratively constructed from, or some combination of these two and generalize the osculating varieties of curves to osculating varieties of arbitrary varieties. Even with this interpretation however, their result is still incomplete as general hypersurfaces with degenerate Gauss maps having fibers of dimension greater than one cannot be built out of cones and osculating varieties. In this note we present examples of varieties with degenerate Gauss mappings. Some of these examples illustrate Girffiths-Harris' structure theorem, and some (see, for example, IIB.) show its incompleteness.Fixing X n ⊂ PV , let r denote the rank of γ and set f = n − r, the dimension of a general fiber. If x ∈ X is a smooth point, we let F = γ −1 γ(x) denote the fiber of γ (which is a P f ). Let Z F ⊆ F ∩ X sing denote the focus of F , the points where the image of a desingularization of X is not immersive. Z F is a codimension one subset of F of degree n − f . The number of components of Z F and the dimension of the varieties each of these components sweeps out as one varies F furnish invariants of X.Here are some examples of varieties with degenerate Gauss mappings (which are not mutually exclusive

We find d − 2 relative differential invariants for a d-web, d ≥ 4, on a two-dimensional manifold and prove that their vanishing is necessary and sufficient for a d-web to be linearizable. If one writes the above invariants in terms of web functions f (x, y) and g 4 (x, y), . . . , g d (x, y), then necessary and sufficient conditions for the linearizabilty of a d-web are two PDEs of the fourth order with respect to f and g 4 , and d − 4 PDEs of the second order with respect to f and g 4 , . . . , g d . For d = 4, this result confirms Blaschke's conjecture on the nature of conditions for the linearizabilty of a 4-web. We also give the Mathematica codes for testing 4-and d-webs (d > 4) for linearizability and examples of their usage.Let W d be a d-web given by d one-parameter foliations of curves on a two-dimensional manifold M 2 . The web W d is linearizable (rectifiable) if it is equivalent to a linear d-web, i.e., to a d-web formed by d one-parameter foliations of straight lines on a projective plane.The problem of finding a criterion of linearizability of webs was posed by Blaschke in the 1920s (see, for example, his book [2], §17 and §42) who claimed that it is hopeless to find such a criterion. Blaschke in [2] formulated the problems of finding conditions for the linearizability of 3-webs ( § 17) and 4-webs ( § 42) given on M 2 . Comparing the numbers of absolute invariants for a general 3-web W 3 (respectively, a general 4-web W 4 ) and a linear 3-web (respectively, a linear 4-web), Blaschke made the conjectures that conditions of linearizability for a 3-web W 3 should consist of four relations for the ninth order web invariants (four PDEs of ninth order) and those for a 4-web W 4 should consist of two relations for the fourth order web invariants (two PDEs of fourth order).A criterion for linearizability is very important in web geometry and in its

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