A bound on the degree of schemes defined by quadratic equations
Projective varieties defined by quadratic equations are a fascinating subject in algebraic geometry by many point of views; here the degree of such a scheme is studied, and a bound for it (in terms of its codimension) is given under a few general condition about syzygies of its ideal.NEWLINENEWLINELet \(X \subset {\mathbb P}^r = {\mathbb P}_{\mathbb C}^r\); a projective scheme of dimension \(n\); we will say that the scheme \(X\) is \textit{reduced in codimension zero} if a general linear space \({\mathbb P}^n\) intersects \(X\) in a reduced scheme, and we will say that \(X\) is \textit{smooth and integral in codimension one} if \(n\geq 1\) and a general linear space \({\mathbb P}^{n-1}\) intersets \(X\) in a smooth integral curve. Suppose that the ideal of \(X\) is generated by quadrics; will say that \(X\) satisfies property \(K_2\) if the trivial (Koszul) relations among those quadrics are generated by linear syzygies.NEWLINENEWLINEThen a formulation of the main result (actually, this is a corollary since the main results uses a condition weaker than \(K_2\)) is the following: Let \(X\subset {\mathbb P}^r\) be an \(n\)-dimensional scheme of degree \(d\) defined by an \(\alpha\)-dimensional linear system \(\Lambda\) of quadrics. Let \(X\) be reduced in codimension 0 and satisfying \(K_2\); if \(c=r-n\geq 2\), then: \(\alpha \geq 2c-2\) and \({d \choose 2}\leq {2c-1\choose c-1}\) (and equality holds iff \(\alpha = 2c-2\)).NEWLINENEWLINEMore can be deduced when equality holds if \(X\) is also smooth and integral in codimension one.NEWLINENEWLINEOne main tool for the proof of this result is the study of apparent double points on the 2-Veronese embedding of a suitable quadric in a linear space \({\mathbb P}^{c}\).NEWLINENEWLINENote that the bound \({d \choose 2}\leq {2c-1\choose c-1}\) is better than the obvious bound \(d\leq 2^c\), and asymptotically is of the form: \(d\leq {2^c\over \sqrt{\sqrt{\pi c}}} .\)
scientific article; zbMATH DE number 6072027