# Tagged Questions

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### Prove $X=(y^{2}z-x^{3}+xz^{2})\backslash\{(1,0,-1)\}$ is irreducible.

Let $X=Z(y^{2}z-x^{3}+xz^{2})\subset\mathbb{P}^{2}$ and $P=(1,0,-1)$. Prove that $X\backslash P$ is irreducible in the Zariski topology. When char$(k)\neq 2$, I've answered this problem by ...
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### Multiplicative Monoid, Ideal, and Cone [on hold]

Let f(x) be polynomial function, i.e. a linear summation of monomials with real constant. Let ${\cal M}$ be Multiplicative Monoid. What is ${\cal M}(f(x))$?
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### Algebraic Geometry and Maximal ideals

I am solving the following problem but couldn't figure out a strategy to solve: Does $(x^3-17, y^2)$ generate maximal ideals in the quotient ring $R=\mathbb{C}[x,y]/I$ where $I$ is the principal ...
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### Nonsingular projective variety of degree $d$

For each $d>0$ and $p=0$ or $p$ prime find a nonsingular curve in $\mathbb{P}^{2}$ of degree $d$. I'm very close just stuck on one small case. If $p\nmid d$ then $x^{d}+y^{d}+z^{d}$ works. If ...
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### Positivstellensatz for non-polynomial

Can we use Positivstellensatz (P-satz) below for a non-polynomial term? P-satz: Let $R$ be real closed field. Let $f,g,h$ be finite families of polynomials in $R[X_{1} ,...,X_{n}]$. Denote by P the ...
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### Hom of algebras

For any $R$-algebras $A$ and $B$, doea their set of R-algebra morphisms $\mathrm{Hom}_{R_{\mathrm{Alg}}}(A,B)$ necessarily again have the strucutre of an $R$-algebra?
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### Example: Irreducible component - affine varieties

Again, I know how to prove the statement. But, I cannot find any example. Please help me for finding an example. Thank you:)
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### Affine variety example.

I know how to show the statement. But I cannot find an example (the part I underlined by a yellow pen) please help me for finding an example. Note: For example, can I consider the following ...
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### Variety and algebraic curves

I am attempting the following problem from Artin: Every variety in $\mathbb{C^2}$ is the union of finitely many points and algebraic curves. I think the proof is trivial (unless I am missing ...
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### Irreducible polynomials and algebraic geometry

I was reading Dummit and Foote and this was one of statements stated (without any proof), "An irreducible curve have finitely many singular points" I would like to know why is this true. Shouldn't it ...
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### Number of common zeros of two quadratic polynomials in ${\Bbb C}[t,x]$

The following theorem is in Artin's Algebra(2nd edition): Theorem 11.9.10 Two nonzero polynomials $f(t,x)$ and $g(t,x)$ in two variables have only finitely many common zeros in ${\Bbb C}^2$, ...
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### Is evaluation homomorphism surjective?

Let $A^n$ be an affine space over $\mathbb{C}$ and let $\mathbb{C}[X_1,\cdots,X_n]$ be the polynomial ring of $n$ variables. Then $A^n\to (\mathbb{C}[X_1,\cdots,X_n])^*$ by evaluation homomorphism, ...
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### The quadratic transformation of $\mathbb{P}^{2}$, $(a_{0},a_{1},a_{2})\mapsto (a_{1}a_{2},a_{0}a_{2},a_{0}a_{1})$.

Let $\varphi:\mathbb{P}^{2}\rightarrow\mathbb{P}^{2}$ defined by $$(a_{0},a_{1},a_{2})\mapsto (a_{1}a_{2},a_{0}a_{2},a_{0}a_{1}).$$ I am tasked to show tha this is a rational map. My goal is to show ...
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### What does $J_1\cap J_2=\emptyset$ mean algebraically for two varieties in $\Bbb{C}^n$?

Let $J_1, J_2$ be two varieties in ${\Bbb C}^n$. Then $$J_i=V(I_i)\quad i=1,2.$$ for some $I_i\subset\Bbb{C}[x_1,\cdots, x_n]$ and $$J_1\cap J_2=V(I_1\cup I_2)$$ and $$J_1\cup J_2=V(I_1I_2).$$ ...
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### What are the ideals in ${\Bbb C}[x,y]$ that contain $f_1,f_2\in{\Bbb C}[x,y]$?

This question is based on an exercise in Artin's Algebra: Which ideals in the polynomial ring $R:={\Bbb C}[x,y]$ contain $f_1=x^2+y^2-5$ and $f_2=xy-2$? Using Hilbert's (weak) nullstellensatz, ...
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### Tensor of quotients question

Let $R = \mathbb{Z}[x]$ be the 1-variable polynomial ring over the the integers, $q$ and $p$ arbitrary prime numbers, $R_1 = \mathbb{Z}[[x]]/(p)$, and $R_2 = \mathbb{Z}[x]/(q, x)$. Question: what ...
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### Isomorphism between polynomial ring and ring of polynomial functions.

Let $A$ be a ring, and $\mathfrak{P}$ denote the ring of polynomial functions on $A$ in $n$ variables (pointwise defined ring operations), i.e. maps $x\mapsto\sum_{\alpha} a_\alpha x^\alpha$ (using ...
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### Flatness question

In reading on the stacks project I came across a result I don't quite follow: "Assume M is finitely presented and flat, i.e., (1) holds. We will prove that (7) holds. Pick any prime p and x1,â€¦,xrâˆˆM ...
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### Question on Nakayama?

In reading a certain proof on the stacks project "http://stacks.math.columbia.edu/tag/00NV", I can't see how Nakayama's lemma is used to make the following conclusion: "Assume M is finitely presented ...
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### Symmetries of a square and it's similarity to the Division Algorithm

I need help with this question: (each variable $r$ represents a rotation of the square about the axis through its centroid at $90^{\circ}$ intervals. $e$ represents nonmotion. This question is ...
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### Invariants of the Determinant Form

Consider a form of degree $r$ in $n$, that is, a homogeneous polynomial $$f(x_1, \ldots, x_n)=\sum_{i_1+\ldots i_n=r}\alpha_{i_1 ... i_n}x_1^{i_1} ... x_n^{i_n}$$ After the linear change of ...