## Unobstructedness of filling secants and the Gruson-Peskine general projection theorem.(English)Zbl 1315.14068

The generic projection conjecture, which predicted that the projection of a smooth embedded projective variety from a generic point has only the expected singularities, has been recently proved by L. Gruson and C. Peskine (see [Duke Math. J. 162, No. 3, 553–578 (2013; Zbl 1262.14058)]). In the paper under review this result is generalized, showing smoothness and results of expected dimension for multiple point loci of generic projections, mainly from a point or a line, or for fibers of embedding dimension 2 or less. As stated in the introduction, three results extending Gruson and Peskine’s result are proved:
Theorem 4.1, where arbitrary ambient spaces are allowed and the result is presented as a general statement about deformation of rational curves on varieties constrained by contact conditions with a fixed subvariety;
Theorem 5.1, where arbitrary-dimension centers of projection are allowed, fibres with local embedding dimension 2;
Theorem 6.1, where arbitrary-centers and contact conditions are allowed, curvilinear fibers.
The approach is different from that of Gruson and Peskine: it encodes the secant or contact conditions in a sheaf $$M$$ (on the secant plane or curve), which controls the corresponding deformations; the geometric hypotheses on these secant or contact conditions imply generic spannedness for $$M$$; since the base is often linear, the generic spannedness implies spannedness and a good control on the deformations.

### MSC:

 14N05 Projective techniques in algebraic geometry

### Keywords:

secants; generic projections; multiple points; rational curves

Zbl 1262.14058

Macnodal
Full Text:

### References:

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