I cannot seem to find stated the following fact, which is surely well known to experts.


Let (S,L) be a polarized K3 surface. Then $M = L^{\otimes 3}$ is very ample and we can consider the embedding in the corresponding projective space $S \to P^N$.


Question: is the image of $S$ in $P^N$ a complete intersection? I suspect the answer is no but I haven't seen it stated explicitly. (by the way, I am certainly allowing $S = H\_1 \cap ... \cap H\_k \cap D\_1 \cap D\_2$, where the $H\_i$ are hyperplanes and $D\_1, D\_2$ or more interesting divisors.


(information about other embeddings of S which are complete intersections, or statements about when one can know if S is a complete interesction would be highly appreciated as well)



are K3 surfaces complete intersections in their polarization?
Suppose $X$ is a smooth surface complete intersection in $\mathbb{P}^n$, given by equations of degrees $d\_1,\ldots ,d\_{n-2}\ $ . I assume $d\_i\geq 2$, because the image of a variety by a complete linear system cannot be contained in a hyperplane. By the adjunction formula the canonical divisor of $X$ is $(\sum d\_i-n-1)$ times the hyperplane section. This is $\,0\,$ if and only if $(d\_1,\ldots ,d\_{n-2})= (4)$, $(2,3)$ or $(2,2,2)$. Thus the only polarizations for which the image can be a complete intersection are those of degree $4$, $6$ and $8$.


