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May
21
comment Zeroes of prime polynomials in the algebraic torus (A Hilbert's Nullstellensatz for Laurent polynomials?)
You are, of course, completely correct. Regarding your second thing, I missed the "different" ... so yea, you are right about that, too.
May
20
comment Can someone illustrate the definition of manifold with a simple example?
The most classical example would be the sphere in $\mathbb R^3$. For any point $x$, you can pick the point opposite to it $y$ and use the stereographic projection map to identify $M\setminus\{y\}$ with $\mathbb R^2$.
May
19
comment Dimension of affine part equals to the dimension of the variety
The dimension of $\mathbb P^n$ is not equal to the Krull dimension of its projective coordinate ring $k[x_0,\ldots,x_n]$, which is $n+1$. That's because $\mathbb P^n$ is not an affine variety. The problem with your question is that it is ill-posed: What is your definition of the dimension of $\mathbb P^n$?
May
18
comment Inclusion of Tori induces surjection of character groups?
This is true, but I have too little time for anything but a reference right now: Tauvel & Yu, Lie algebras and algebraic groups, 22.5.4 (iii) is a slightly more general statement for diagonalizable algebraic groups. Any torus is diagonalizable, of course.
May
12
comment Inclusionwise maximal linear subvarieties of a projective variety
@DanielMcLaury: Interesting, I didn't see that angle. If I am not somehow mistaken, this is the image of a projective morphism, so it should be closed. Any flaws in that argument?
May
6
comment what is the precise definition of a morphism defined over $k$?
@ZhenLin: That's true, but along with your comment above this should be enough for the OP, I hope.
May
4
comment A confusion on the definition of morphism between varieties
I believe the answer to your question might be Lemma I.3.6 in Hartshorne.
Apr
22
comment Every variety contains open affine normal subvariety
@DanielMcLaury: Good point, fixed.
Apr
21
comment Is the preimage of the non-normal locus a divisor?
@AsalBeagDubh: A very good nitpick, I fixed that.
Apr
20
comment Multiplicity of point as a zero of polynomial.
Also, how do you define $I_p(C,L)$?
Mar
19
comment Are unipotent algebraic groups connected?
this answer on mathoverflow strongly suggests that your argument is solid.
Mar
16
comment An isomorphism from $GL_{2}(\mathbb{F_{2}})$ to $S_{3}$
These two groups are not isomorphic. $\operatorname{GL}_3(\mathbb F_2)$ has at least $7$ elements, because there are $7$ nonzero vectors in $\mathbb F_2^3$. On the other hand, $S_3$ only contains $6$ elements.
Mar
11
comment Resolution of singularities of the determinant hypersurface
This is indeed a very nice resolution. I would have preferred an embedded one, but this is quite a nice start, so +1 and accept. Thanks!
Mar
11
comment Can a birational morphism surject from an affine to a projective variety?
+1 and accept, also thank you, this is a nice proof.
Mar
7
comment True or False: $f$ is injective if and only if $f^*$ is surjective where $f^*$ is corresponding to the pullback to $f$.
Of course, you can easily turn this into a direct proof, which is then more elegant. But it is late, and I will leave you with this.
Mar
6
comment Can a birational morphism surject from an affine to a projective variety?
@AlexYoucis: What is the argument for the fact that it has an inverse on some $U\subseteq Y$ whose complement is codim. 2? The approach sounds interesting!
Mar
4
comment Can a birational morphism surject from an affine to a projective variety?
@karl_christ: No, I mean surjective. Every fiber is nonempty.
Mar
4
comment Can a birational morphism surject from an affine to a projective variety?
@karl_christ: Such a map would only give me an isomorphism of a dense open subset $U\subseteq X$ with another dense open subset $V\subseteq Y$ of $Y$. This proves nothing.
Mar
4
comment Can a birational morphism surject from an affine to a projective variety?
@KReiser: Yea, I edited my question, this only makes sense for irreducible varieties.
Mar
3
comment Can a birational morphism surject from an affine to a projective variety?
@Ben: I am not so much interested in pathologic cases, you may safely assume that all the objects are actually interesting.