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:
“Every gazette brings accounts of the untutored freaks of the wind,—shipwrecks and hurricanes which the mariner and planter accept as special or general providences; but they touch our consciences, they remind us of our sins. Another deluge would disgrace mankind.”
—Henry David Thoreau (1817–1862)
“The word “jealousy” is often used as if it were synonymous with envy; but I think the distinction worth preserving. Jealousy is predominantly concerned with the fear of loss of something one possesses, envy with the wish to own something another possesses. Othello suffers from the fear that he has lost Desdemona’s love. Iago suffers from envy of the position held by Cassio, to which he feels entitled.”
—Anthony Storr (b. 1920)