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:
“We all have known
Good critics, who have stamped out poet’s hopes;
Good statesmen, who pulled ruin on the state;
Good patriots, who, for a theory, risked a cause;
Good kings, who disembowelled for a tax;
Good Popes, who brought all good to jeopardy;
Good Christians, who sat still in easy-chairs;
And damned the general world for standing up.—
Now, may the good God pardon all good men!”
—Elizabeth Barrett Browning (1806–1861)
“The essence of spirit, he thought to himself, was to choose the thing which did not better one’s position but made it more perilous. That was why the world he knew was poor, for it insisted morality and caution were identical.”
—Norman Mailer (b. 1923)