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I have learnt that every projective variety has a (unique) decomposition into irreducible components. But I have not decomposed any particular projective varieties. Now my homework includes a few problems of computing the irreducible components.

Here is one of them: decompose $V(X_2^2-X_1X_3, X_0X_3^2-X_2^3)\subset \mathbb{P}^4$. I would like to see how this is done as an example so that I might be able to solve the rest myself after seeing how this one is done. Also are there any general strategies?

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Have you tried showing your variety is irreducible? You could use the "dictionary" between closed subsets of projective $n$-space and ideals of k[a_0,\ldots,a_n]$. Being irreducible as a closed subset usually boils down to the polynomials being irreducible. – Harry Feb 3 '13 at 21:06
This one is somewhat difficult. You have probably seen the solution by now, but if not, I'll just quickly say that one component is the line $V(X_2, X_3)$, and the other is the twisted cubic. (Also, the variety is supposed to be considered in $\mathbb{P}^3$.) – Zhen Lin Feb 18 '13 at 23:17

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