multiset) of the degrees of defining hypersurfaces. For example, taking quadrics in P3 again, (2,2) is the multidegree of the complete intersection of two of them, which when they are in general position is an elliptic curve. The Hodge numbers of complex smooth complete intersections were worked out by Kunihiko Kodaira. General position For more refined questions, the nature of the intersection has to be addressed more closely. The hypersurfaces may be required to satisfy a transversality condition (like their tangent spaces being in general position at intersection points). The intersection may be scheme-theoretic, in other words here the homogeneous ideal generated by the Fi(X0, ..., Xn) may be required to be the defining ideal of V, and not just have the correct radical. In commutative algebra, the complete intersection condition is translated into regular sequence terms, allowing the definition of local complete intersection, or after some localization an ideal has defining regular sequences. Topology Homology Since complete intersections of dimension in are the intersection of hyperplane sections, we can use the Lefschetz hyperplane theorem to deduce that for . In addition, it can be checked that the homology groups are always torsion-free using the universal coefficient theorem. This implies that the middle homology group is determined by the Euler characteristic of the space. Euler characteristic Hirzebruch gave a generating function computing the dimension of all complete intersections of multi-degree . It reads Citation References External links Complete intersections at the Manifold Atlas Algebraic geometry Commutative algebra