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I have two very basic questions/clarifications.

Let $X=\mathbb{P}^n_k$, and let $Y$ be a subvariety of $\mathbb{P}^n_k$ of dimension $m$. Then we say that $V$ is a complete intersection variety if there exists $n-m$ polynomials $F_1,\cdots,F_{n-m}$ such that the ideal of $V$, $\mathcal{I}(V)$ is generated by $F_i$-s : $1\leq i\leq n-m$.

1) Now, if $X=\mathbb{P}^2_k$, any curve $C$ is a complete intersection subvariety right? Because any curve is a subvariety of dimension 1 and is the zero set of a single homogeneous polynomial.

2) If $X$ is any surface (smooth, projective variety over $k$ of dimension 2). Again any curve $C$ is a subvariety of dimension 1 and it is the zero set of a single section $s\in H^0(X,\mathcal{O}_X(1))$. So again any curve in any surface is a complete intersection subvariety?

Thanks in advance!

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1) That is correct. 2) A few things. First, there is no intrinsic $\mathcal O_X(1)$, and once the surface is embedded, then clearly not every curve is a hyperplane section of the surface (this isn't even true in $\mathbb P^2$ for instance). More important, though, is that not every curve is even a candidate. For example, a single ruling on a smooth quadric surface in $\mathbb P^3$ cannot be defined by a single equation. (It is true that a curve on a smooth surface is a local complete intersection, meaning that at any point there is a neighborhood of that point for which the curve is defined by one equation.)

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