The theorem of Mather on generic projections in the setting of algebraic geometry (Q1191444)

A theorem by \textit{J. Mather} [Ann. Math., II. Ser. 98, 226-245 (1973; Zbl 0267.58005)] on generic projections has been applied by the authors of this note in the work around the Hartshorne conjecture about the projective varieties of small codimension. Mather's proof requires a careful local study of the Thom-Boardman singularities in the jet space and it is not easy to read for a non-specialist in the field. C. Peskine encouraged us to write down a self-contained account of Mather's proof in the setting of (complex) algebraic geometry and this is exactly what we try to do in this note: Let \(X\) be a smooth subvariety of the complex projective space \(\mathbb{P}^ n\) of codimension \(d\). Let \(T\) be any linear subspace of \(\mathbb{P}^ n\) of dimension \(t\) such that \(T \cap X=\emptyset\) (so \(t<d)\). For any \(i_ 1 \leq t+1\) let \(X_{i_ 1}=\{x \in X | \dim [T_ x X_ 1 \cap T]=i_ 1-1\}\). When \(X_{i_ 1}\) is smooth for any \(i_ 2 \leq i_ 1\) define \(X_{i_ 2,i_ 2}=\{x \in X_{i_ 1} | \dim [T_ xX_{i_ 1} \cap T]=i_ 2-1\}\) and so on, for \(i_ 1\geq i_ 2 \geq \cdots \geq i_ k\) define (when possible) \(X_{i_ 1, \dots, i_ k}\). For \(T\) in a Zariski open set of the grassmannian \(\text{Gr} (\mathbb{P}^ t, \mathbb{P}^ n)\) we have that each \(X_{i_ 1, \dots, i_ k}\) is smooth (and so the definitions above are possible) until (increasing \(k)\) it becomes empty and its codimension in \(X_ 1\) is equal to the number \(\nu_{i_ 1, \dots, i_ k,0, \dots,0}\) defined below. Our feeling is that Mather's theorem is promising for further applications in algebraic geometry.