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It must be set-theoretically equidimensional, but not scheme-theoretically. (Consider two lines in space colliding, developing an embedded point at the intersection.) For the positive statement, let …

Obviously $\widetilde{B_1} \cap \widetilde{B_2} \subseteq \widetilde{B_1\cap B_2}$. I'll discuss a sufficient condition for the reverse. If $B_1\cap B_2$ is equidimensional, then so is $\widetilde{B_ …

Equivalently, does $C$ have a subscheme $C'$ such that $p_C - p_{C'}$ is finite? $C$ satisfies Serre's condition S1 iff $C$ is the closure of the union of the generic points of its geometric componen …

I think more fundamental and interesting than the notion of "multiplicity" is the notion of "tangent cone". Given $x \in X = Spec \ R$ defined by an ideal $I$, it's an interesting fact that $I^\infty$ …

(3) is nonsensical. For schemes of dimension $>0$, CM implies S1 which means no embedded components, like you were suggesting $X\cap Y$ to have. (4) is true. A subscheme of affine space is a union of …

Exactly as Alexander said, this will fail for $G/P$ nonminuscule. The smallest example is the closed orbit $SO(5)/P$ of $SO(5)$ acting on ${\mathbb P}({\mathbb C}^4)$. The $T$-weight diagram of this r …

Let $R$ be a commutative Noetherian $F$-algebra, where $F$ is a field (perfect, say). Assume that $R \otimes_F \overline F$ is a polynomial ring over the algebraic closure $\overline F$. Does it fo …

Let $R$ be a finitely generated commutative ring over a field, for concreteness. If $S,T \leq R$ are two finitely generated subrings, is their intersection also finitely generated? (Certainly …

I give some hints about computing them in my paper Balanced normal cones and Fulton-MacPherson's intersection theory, section 3. (Also I compute 14 examples.) I find it kind of astounding that this e …

There's an exercise in [Eisenbud] that says that reducedness is R0 + S1, i.e. generic reducedness plus Serre's condition S1, which says that there are no embedded primes. This is analogous to normalit …

It's kinda gross, but it can be done. To each monomial, add a generic linear combination of all smaller monomials (w.r.t. your term order). Now insist that what you have is a Gr\"obner basis. How do …

OOPS: As Sándor points out, I missed the assumption that the ring is local, in the following example of the wrong thing: "Inside 3-space, glue together a plane $y=0$ transversely with a parabola $z=0 …

Computing colon ideals is pretty quick. You could colon out the variables in order. If the ideal changes, record the variable that worked, and go back to the beginning of the list. Either you get to t …

I'd like to see a complete proof of the simplest version of the following rough statement: "If $f/g$ is a rational function on a reduced scheme ($g$ not a zero divisor), and $f/g$ doesn't have poles i …

In [Hartshorne, III.3] he proves that injective modules over $R$ give flasque sheaves over $Spec\ R$. I presume that's because they don't give injective sheaves, and flasque is the consolation prize. …