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Let $k$ be a field of characteristic zero (non necessarily algebraically closed, we may assume for instance that $k = \mathbb{C}(t)$). Does there exist a classification of degree four surfaces $S\subset\mathbb{P}^3_{k}$ (defined over $k$) up to the action of $PGL(4,k)$ or of the Cremona group of $\mathbb{P}^3_{k}$?

Thank you.

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    $\begingroup$ By the adjunction formula these are K3 surfaces, so you might want to look at the literature on them. $\endgroup$
    – Tabes Bridges
    Commented Dec 29, 2022 at 0:34
  • $\begingroup$ I do not think they are $K3$ in general. For instance, a quartic surface with a double line is not $K3$. $\endgroup$
    – Puzzled
    Commented Dec 29, 2022 at 13:12

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A fine classification of the quartic surfaces that are not normal is in

Tohsuke URABE, "Classification of Non-normal Quartic Surfaces", TOKYO J. MATH. VOL. 9, No. 2, 1986, 265-295

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    $\begingroup$ Yes, I know this paper but I think this classification works just when the base field is algebraically closed. $\endgroup$
    – Puzzled
    Commented Dec 29, 2022 at 17:58

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