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Here the base field is the field of complex numbers.

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Maybe add the "k3-surfaces" tag? –  Noam D. Elkies Jun 11 '11 at 14:01

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Any $K3$ surface $S$ of degree $8$ in $\mathbb{P}^5$ is contained in $3$ linearly independent quadrics. It can be seen that in the general case $S$ is a complete intersection of $3$ quadrics. However, there are some special cases where this is not true, but they can be completely described.

The point is that $S$ is a complete intersection of $3$ quadrics if and only if a general hyperplane section (which is a smooth curve of genus $5$) has no $g_3^1$.

You can find all details in [Beauville, Complex Algebraic Surfaces, Chapter VIII, Example 14 and Exercise 11].

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