On the Chern classes and the Euler characteristic for nonsingular complete intersections

Abstract

It is shown that Hirzebruch’s result on the Chern classes of a complete intersection of nonsingular hypersurfaces in general position in P N ( C ) {{\mathbf {P}}_N}({\mathbf {C}}) , is also valid for any nonsingular complete intersection. Then relations between Euler characteristic, class and Milnor number are pointed out.

Rational points on the intersection of three quadrics

“…In our case Lemma 2.2 implies in particular that there are linear combinations m (1) .Q and m (2) .Q , with m (1) , m (2) ∈ K 3 , such that both the hypersurface m (1) .Q = 0 and the intersection m (1) .Q = m (2) .Q = 0 are nonsingular. In particular we will have d 1 (m (1) ) = 0, so that the form d 1 (t) does not vanish identically.…”

“…In particular we will have d 1 (m (1) ) = 0, so that the form d 1 (t) does not vanish identically. Moreover it follows from Lemma 2.1 that the intersection m (1) .Q = m (2) .Q = 0 is nonsingular if and only if d 2 (m (1) , m (2) ) is non-zero, and so we deduce that the form d 2 (t (1) , t (2) ) does not vanish identically.…”

“…We also define the determinant δ(X, Y ; t (1) , t (2) ) := det(Xt (1) .Q + Y t (2) .Q ) and the discriminant d 2 (t (1) , t (2) ) := Disc(δ(X, Y ; t (1) , t (2) )).…”

“…Thus δ(X, Y ; t (1) , t (2) ) is a form of degree n in X and Y , and d 2 (t (1) , t (2) ) is bihomogeneous in the entries of t (1) and t (2) .…”