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Apr
6
comment Harris' AG ex 2.24: projective variety under regular map.
First of all, thank you so much for your help. Second, Which book do you recommend for an absolute beginner? Lots of books seem to offer pretty convoluted definitions for concepts that I imagine have rich intuitive meaning and a lot of the beauty seems to be getting lost in commutative algebra (which I also have never studied).
Apr
6
accepted Harris' AG ex 2.24: projective variety under regular map.
Apr
5
revised Harris' AG ex 2.24: projective variety under regular map.
edited title
Apr
5
revised Harris' AG ex 2.24: projective variety under regular map.
added 4 characters in body
Apr
5
comment Harris' AG ex 2.24: projective variety under regular map.
And defining it as $x_{n+i}^d-\phi_i$ would clearly not work, since our points on the graph don't satisfy this polynomial. I just cannot think of a way to get homogenous polynomials that would just carve out the image of the regular map. That is, is the bigger graph where we are just looking at the regular map between the two projective spaces?
Apr
5
comment Harris' AG ex 2.24: projective variety under regular map.
Sure sure. Restricting our attention to just the $\phi_i$ being homogenous and of the same degree, say $d$, how should I define the polynomials, since I cannot just use $x_{n+i}-\phi_i$ (as I would in the affine case) since this is not homogenous.
Apr
5
accepted Quadratic Extension of Finite field
Apr
5
accepted A linear subspace of projective space
Apr
5
accepted Is primitivity invariant under matrix conjugation.
Apr
5
accepted Show that $k[x,y,z]/(xz-y^2)$ is not a UFD.
Apr
5
revised Harris' AG ex 2.24: projective variety under regular map.
added 17 characters in body
Apr
5
comment Harris' AG ex 2.24: projective variety under regular map.
I am not saying $\phi_i$ is a regular map, but $\phi=[\phi_0:\ldots:\phi_i:\ldots:\phi_k]$ is a regular map, right? I completely believe you that the machinery of schemes is useful here, but I don't think it's necessary given that it's given in chapter 2 of Harris book.
Apr
5
comment Harris' AG ex 2.24: projective variety under regular map.
How is that identity not of that form? Isn't the identity just that with $\phi_i([X_0:\ldots:X_k])=X_i$? If I were to assume all regular maps were of this form where each $\phi_i$ is homogenous, do you have any advice for the question? I agree with you that this form of a regular map does not cover all of them, as I've read in Harris's book.
Apr
5
asked Harris' AG ex 2.24: projective variety under regular map.
Apr
5
comment Given $\sum |a_n|^2$ converges and $a_n \neq -1$, show that $\prod (1+a_n)$ converges to a non-zero limit implies $\sum a_n$ converges.
It may very well be, my teacher just put it on the problem set. He took problems from lots of different books without citing them, sometimes because he made minor changes.
Apr
3
revised Show that $k[x,y,z]/(xz-y^2)$ is not a UFD.
deleted 4 characters in body
Apr
3
asked Show that $k[x,y,z]/(xz-y^2)$ is not a UFD.
Mar
31
awarded  Civic Duty
Mar
31
comment Problems with votes
please complete the question
Mar
31
accepted Polynomials vanishing and non-intersecting varieties.