The theorem of Bertini on the variable singular points of a linear system of varieties

Pure transcendental extensions of the ground field. Let V/k be an irreducible r-dimensional algebraic variety over a given ground field k. We assume that V/k is immersed in an n-dimensional projective space and we denote by x1, X2, * * *, x, the nonhomogeneous coordinates of the general point of V/k. Let u1, U2, * *, Um be indeterminates with respect to the field k(x) [=k(xl, X2, , * * xn)] of rational functions on V/k. We adjoin these indeterminates to the field k(x) and we denote by K the field k(u) [=k(ui, U2, * , urn)]. This subfield K of the field k(x, u) we take as new ground field, and over this new ground field we consider the variety V/K defined by the same general point (x1, X2, * * *, x,) as V/k. The varieties V/k and V/K are of the same dimension r over their respective ground fields k and K. We shall say that the variety V/K is the extension of the variety V/k under the ground field extension k--K. By precisely the same argument, every irreducible subvariety W/k of V/k has as extension an irreducible subvariety W/K of V/K, of the same dimension as 147/k. If xl, x2, * , xn are the nonhomogeneous coordinates of the general point of W/k, then xl, x2, * *, xn are also the nonhomogeneous coordinates of the general point of W/K. Moreover, u1, U2, * * *, urn are indeterminates with respect to the field k(x), that is, they are algebraically independent over this field. Not every irreducible subvariety of V/K is the extension of a subvariety of V/k, but every such subvariety W*/K defines an irreducible subvariety W/k of V/k, which we shall refer to as the contraction of W*/K and which is obtained as follows. Let xl*, x2, , x* be the nonhomogeneous coordinates of the general point of W*/K. Since W*/KC V/K, the ring K [xl*, x2, Xn*] is a homomorphic image of the ring K [xl, X2, , xn. Therefore, also the ring k [x1*, X2*, , x* is a homomorphic image of the ring k [xl, X2, Xn]. Therefore there is a unique irreducible subvariety W/k of V/k, whose general point (x1, x2, * , n) is defined by the condition that the rings k[x1, X2, * **, x~] and k[xe, x, , x*] be simply isomorphic and that xi, xi* (i= 1, 2, , n) be corresponding elements in the isomorphism. This variety W/k shall be termed the contraction of W*/K.