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I started learning algebraic geometry, and I still can grasp how to answer computational questions.

I have a few questions that I don't know how to start.

one of them:

Let Y = Z($x^2-z^3+1, y^2-w^3-w-1)\subseteq \mathbb{A}^4$:

a. Prove that Y is an affine variety.

b. Find $dim(Y)$

c. is Y differentiable?

d. find a projective closure for Y

e. find a hyper surface $S\subseteq \mathbb{A}^{\text{dim}(Y)+1}$ so that Y and S are birational.

I'm still having problems with a. I know that it is an algebraic set (and closed in the Zariski topology). I don't know how to prove that it is irreducible (I don't know how to prove that I(Y) is prime or that $K[x,y,z,w]/I(Y)$ is an integral domain.

john
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Nir Agami
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  • Note that $k[x,y,z,w]$ is a UFD. Can you prove that the two polynomials defining the equations are both irreducible? Then $(k[x,z]/(x^2-z^3+1))[y,w]$ is a domain. Can you proceed further from here? – random123 Aug 22 '21 at 19:04

1 Answers1

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Let $X_1: x^2 = z^3 - 1$ and $X_2: y^2 = w^3 + w + 1$, which are curves in $\mathbb{A}^2$. Since $z^3 - 1$ and $w^3 + w + 1$ are squarefree, assuming the characteristic of the base field is not $3$ or $31$, then $X_1$ and $X_2$ are irreducible by Eisenstein's criterion. Then $Y = X_1 \times X_2$, and since the product of irreducible varieties is irreducible (see here or here), then $Y$ is irreducible.

Viktor Vaughn
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