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The following excerpt is from page 147 of Dieudonne's History of Algebraic Geometry:

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Can anyone provide a reference for this result? Is it difficult to find explicit equations for the two hypersurfaces? For example: I would like to find two equations for the curve of genus $2$ in $\mathbb{CP}^3$ defined by the following $3$ polynomials:

\begin{align} F &= z_{11}z_{22}-z_{12}z_{21}\\ G &= z_{11}^3+2 z_{21}^2z_{22}+z_{11}z_{12}^2\\ H &= z_{12}^3+2z_{21}z_{22}^2+z_{11}^2z_{12} \end{align}

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    $\begingroup$ The statement about codimension 2 subvarieties doesn't make sense: it would imply for instance that every non-degenerate smooth codimension 2 subvariety of $\mathbb{P}^6$ is a complete intersection — a well-known open problem. In particular, there is no way of writing your genus 2 curve as a complete intersection — see for instance this MO question. $\endgroup$
    – abx
    Commented May 11, 2021 at 13:43
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    $\begingroup$ I believe that Dieudonne is speaking about set-theoretic complete intersection. $\endgroup$
    – Drew Armstrong
    Commented May 11, 2021 at 14:13
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    $\begingroup$ Oh, I see — quite possible. But still: his statement would imply that every curve in $\Bbb{P}^3$ is a set-theoretic complete intersection, again a well-known open problem. I don't know if this is known for a genus 2 curve. $\endgroup$
    – abx
    Commented May 11, 2021 at 14:27
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    $\begingroup$ Also, the Horrocks-Mumford abelian surface in $\Bbb{P}^4$ is a counter-example. $\endgroup$
    – abx
    Commented May 11, 2021 at 14:36

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There are possibly a few confusions in Dieudonné's statement. Here what is known in the direction of the famous Hartschorne conjecture (see also this question and the answers below):

$\bullet$ : If $X \subset \mathbb{P}^n$ is smooth and defined, scheme-theoretically, by less than $\frac{n}{2}$ equations, then it is a complete intersection. This is a result of Faltings, which has been then improved by Netsvetaev.

$\bullet$ Let $X \subset \mathbb{P}^n$ is smooth of codimension $2$ and $n \geq 6$. If $\deg(X) < (n-1)(n+5)$, then $X$ is a complete intersection. This is a result of Holme and Schneider building on previous work by many authors, including Barth-VandeVen, Ran and Chiantini-Ballico. The paper by Holme and Schneider is very well-written and is a nice survey of the state of the art back in 1985.

It is well possible that Dieudonné was thinking to Falting's result and made typos writting degree instead of number of equations and $\geq$ instead of $\leq$.

Edit : By the way, these results do not give, in general, explicit equations. It seems however that the ones by Faltings and Netsvetaev show that given a set of equations defining scheme-theoretically $X$ with the right cardinality (as in their theorems), then equations can be choosen in this set so that they define scheme-theoretically $X$ as a complete intersection.

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