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comment For any smooth n-manifold $M$, construct a smooth map $f:M\to S^n$ which is not null-homotopic
This is certainly impossible for $M=\mathbb R^n$
Jul
1
answered Which is the correct definition of degree of a curve
Jul
1
awarded  Revival
Jul
1
answered Examples of base points of linear systems
Jul
1
comment Is it true that for every $n\in\mathbb{N}$ there exists an algebraic curve $C$ and a point $p$ in that curve whose tangent plane has dimension $n$?
Maybe you could give a hint for the proof that your curve has an $n$-dimensional tangent space at the origin?
Jul
1
answered Is it true that for every $n\in\mathbb{N}$ there exists an algebraic curve $C$ and a point $p$ in that curve whose tangent plane has dimension $n$?
Jul
1
comment Is it true that for every $n\in\mathbb{N}$ there exists an algebraic curve $C$ and a point $p$ in that curve whose tangent plane has dimension $n$?
You should replace the expression "tangent plane" by "tangent space" since a plane necessarily has dimension two.
Jul
1
comment Difference between quadric and conic
A conic is a one-dimensional quadric. End of story.
Jun
30
comment On a scale of 1 to 10 how far is this manifold from being a normal bundle?
@Saal: that's an excellent question. Every submanifold $Y\subset M$ has a very intrinsic normal bundle $ \nu(Y)=\frac {T(M)|Y}{T(Y)}$ which only depends on the humble (hum, hum,...) differential structures on $M$ and $Y$. Whenever you put a Riemannian structure $g$ on $M$ you get a "concrete" normal bundle $N^g(Y)\subset T(M)|Y$ on $Y$. That all these concrete normal bundles are isomorphic when $g$ runs through the zillion Riemannian structures on $M$ is indeed an astonishing fact, best explained by their being all isomorphic to the intrinsic $\nu(Y)$.
Jun
30
comment On a scale of 1 to 10 how far is this manifold from being a normal bundle?
+1, also for your refereshingly original and humorous style.
Jun
30
reviewed Approve What is the difference between submodules of $A/\mathfrak a$ as an $A$-module or as an $A/\mathfrak a$-module?
Jun
30
comment Example of $Q((x))$ that doesnt match field of fractions of ring $F[[x]]$
@rschwieb: The OP said he couldn't come up with "any simple-enough counter-example of this $Q$ and $F$ " and I gave a very explicit one. If he or anybody else has some specific question I'll be glad to address it but since this might be homework I preferred not to give all the details right now.
Jun
29
answered Example of $Q((x))$ that doesnt match field of fractions of ring $F[[x]]$
Jun
29
comment Definition of regular functions on a projective variety
Yes, exactly ${}{}$.
Jun
29
comment Definition of regular functions on a projective variety
By the way, you are right: an open subset of a quasi-projective variety is quasi-projective.
Jun
29
comment Definition of regular functions on a projective variety
He does not define regular functions on quasi-projective varieties on page 18: he defines regular functions on affine varieties $X\subset \mathbb A^n$. Anyway, I strongly advise you not to try to study basic algebraic geometry from that book.
Jun
29
comment How to show that a map is finite
(follow-up of previous comment) Much more generally Bézout's theorem is a wonderful example of a theorem valid in projective space and false in affine space. It is quite normal and healthy to begin with affine varieties. But then one must move on to projective space where many more interesting theorems can be proved. This is similar to the process of first learning differential calculus in open subsets of $\mathbb R^n$ and then moving on to differential manifolds.
Jun
29
comment How to show that a map is finite
Dear aGer, I see them because, like many algebraic geometers, I'm used to working essentially in projective space. You see, the basic definitions, both in classical algebraic geometry and scheme theory, are given first in affine space because there the algebra is nice and easy. However one soon becomes aware of the deficiencies of affine space: for example two distinct lines in an affine plane can meet in two points or one (if they are parallel). This doesn't happen in a projectiveplane where they always meet in exactly one point.