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.
Read more about this topic: Complete Intersection
Famous quotes containing the words general and/or position:
“Each victim of suicide gives his act a personal stamp which expresses his temperament, the special conditions in which he is involved, and which, consequently, cannot be explained by the social and general causes of the phenomenon.”
—Emile Durkheim (1858–1917)
“Einstein is loved because he is gentle, respected because he is wise. Relativity being not for most of us, we elevate its author to a position somewhere between Edison, who gave us a tangible gleam, and God, who gave us the difficult dark and the hope of penetrating it.”
—E.B. (Elwyn Brooks)