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Tangents and secants of algebraic varieties. (English) Zbl 0795.14018
Translations of Mathematical Monographs. 127. Providence, RI: American Mathematical Society (AMS). vii, 164 p. (1993).
In this well written monograph, the author investigates several concrete properties of projective varieties. While a number of results are proved in a very general context, the focus is on nonsingular projective varieties of small codimension. Some conjectures of Hartshorne concerning such varieties are also proved.
The main technical results are proved in a relative setup where one considers a pair of a variety and a subvariety, and the generality in which these results are proved allows the variety to have arbitrary singularities. The principal technique is a detailed study of tangent spaces, secant varieties and higher secant varieties. – Some analogous results are also obtained for subvarieties of complex tori, and certain projective varieties which arise as orbits of highest vectors of irreducible representations of a semisimple Lie group. The latter are designated as $$HV$$-varieties. Explicit birational isomorphisms are constructed between $$HV$$-varieties and projective spaces of the appropriate dimension. $$HV$$-varieties and these birational isomorphisms play a role in proving results for more general projective varieties.
To give a more precise description of some of the results, consider the following (simplified) situation: $$X$$ is an irreducible nonsingular variety of dimension $$n$$ embedded in the projective space $$\mathbb{P}^ N$$ over an algebraically closed field of characteristic zero, $$X$$ is nondegenerate (i.e. not contained in any hyperplane), and $$Y$$ is an irreducible variety of $$X$$ of dimension $$r$$. Here is a sample of the results proved in this situation:
(1) $$X$$ can be isomorphically projected to a projective space $$\mathbb{P}^ M$$ with $$M<2n$$ if and only if it can be projected to $$\mathbb{P}^ M$$ without ramification.
(2) (Theorem on tangencies). If an $$m$$-dimensional linear subspace $$L$$ of $$\mathbb{P}^ M$$ is tangent to $$X$$ along $$Y$$ (i.e. $$L$$ contains $$T_{\chi_ y}$$, the (embedded) tangent space to $$X$$ at $$y$$, for all $$y\in Y$$) then $$r\leq m-n$$.
(3) If $$N<2n$$ then all hyperplane sections (not only the general ones as asserted by Bertini) of $$X$$ are reduced, whereas if $$N<2n-1$$ then all hyperplane sections are normal.
(4) $$X$$ can be isomorphically projected to $$\mathbb{P}^{N-1}$$ if and only if for a general (resp. each) pair of points of $$X$$ there exists a hyperplane that is tangent to $$X$$ at these points.
(5) If $$X$$ can be projected isomorphically to $$\mathbb{P}^ M$$ with $$M<N$$ then $$3N\leq 2(M-1)$$.
(6) If $$3n>2(N-1)$$ then $$X$$ is linearly normal, i.e. the linear system of hyperplane sections of $$X$$ is complete.
(7) $$X$$ is called a Severi variety if $$3n=2(N-2)$$ and $$X$$ can be isomorphically projected to $$\mathbb{P}^{N-1}$$. Severi varieties are classified by showing that each Severi variety is an $$HV$$-variety and that there exist exactly four of these and they have dimensions 2, 4, 8 and 16.
(8) For a positive integer $$k$$, the higher secant variety $$S^ k X$$ is by definition the closure of the union of $$k$$-dimensional linear subspaces spanned by generic collections of $$k+1$$ points of $$X$$. If $$X$$ spans the whole of $$\mathbb{P}^ N$$ then $$S^ k X= \mathbb{P}^ N$$ with $$k=[n/\delta]$$, where $$\delta=2n- \dim(S^ 1 X)+1$$. Consequently, for a given $$n$$ and $$r\leq 2n$$ there is a bound for the maximal number $$N$$ for which there exists an $$n$$-dimensional $$X$$ contained in $$\mathbb{P}^ N$$ that can be isomorphically projected to $$\mathbb{P}^ r$$. This bound is sharp; the varieties for which it is attained are called Scorza varieties (a Severi variety is a special case of a Scorza variety with $$\delta= n/2$$). It is shown that each Scorza variety is an $$HV$$-variety, and these are completely classified.
Reviewer: B.Singh (Bombay)

##### MSC:
 14J10 Families, moduli, classification: algebraic theory 14M07 Low codimension problems in algebraic geometry 14N05 Projective techniques in algebraic geometry 14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry 14Jxx Surfaces and higher-dimensional varieties