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.

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 |