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Given an affine scheme $S=\operatorname {Spec (R)} $ and a closed subscheme $T=V(J)\subset S$, the restriction mapping $$\mathcal O(S)=R\to \mathcal O(T)=R/J$$ is obviously surjective. Applying this t …

Given an exact sequence of locally free sheaves $0\to E'\to E\to E''\to 0$ on, say, a ringed space, there is a canonical isomorphism $\text {det} E=\text {det} E'\otimes \text {det} E''$: this is pur …

Hochster has an elegant construction which associates a commutative ring to each infinite totally ordered set with the property that strictly between two distinct elements there is a third one. The …

Yes, it is true: the analytic and algebraic closures of $X$ in $Z$ coincide (and you don't need at all to assume that $Z$ is smooth). You may suppose that X is open in $Z$: if it isn't, just replace …

Dear Randy, here is an easy to remember, terse slogan: Normal does not imply non-singular, except in dimension one, where it does. (Details in Hartshorne, Atiyah-MacDonald Prop.9.2 and all the other …

The short (and unfriendly!) answer is that it is an irreducible component of the Hilbert scheme. Here is a slightly more detailed answer. Given a subvariety $X\subset \mathbb P^N$, you can attach to …

There is an advanced monograph by F.L.ZAK on exactly this subject: "Tangents and secants of algebraic Varieties" (AMS, 1993, Translations of Mathematical Monographs 127). A free pdf version is online …

The explanation why a lot of the algebraic data is encoded in lower dimension might reside partially in the following theorem. Let X/k be smooth and irreducible over the field k. Let $F \subset X $ b …

As with all definitions, there is no "proof" that the adopted definition is the right one but only a feeling that it better corresponds to our intuition. In the case at hand, taking $R/IJ$ as structu …

Dear mingming, you can find a lot of information on secant varieties in Harris's book "Algebraic Geometry, A First Course "(Springer GTM 133), essentially presented as a set of thoughtfully conceived …

Dear Undergraduate Student, first of all, congratulations on the beautiful geometry you chose to study so early in your studies. To prove $dim X_3 \leq 2$ it is indeed enough to prove that there exis …

Dear Workitout, of course I can't prove there is no link, but I'm rather pessimistic . Here is a fuzzy argument in support of my feeling. The set of sections of $\mathcal O_X$ is never empty (after a …

Dear Robert, the joke on page 91 is that the ruled quadric depicted has "Central Electricity" written over it. It is an allusion to the cooling towers used by power plants. Here is the obligatory Wiki …

Dear Damien, let's show that your morphism $f: \mathbb A^1_K \to \mathbb A^n_K $ is proper, hence closed, hence certainly has closed image. For that it is enough to prove that each $f_i:\mathbb A^1 …

Dear fishibones, an important point is that at the beginning of Chapter II, Bourbaki states that all the rings he will consider are commutative. This implies that if a free module has finite dimension …