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The literature about K3 surfaces is extensive. Let us consider the fact that the zero locus of any smooth homogeneous degree 4 equation is a K3 surface.

Let $R$ the quotient ring of a homogeneous polynomial in $4$ variables and degree $4$.

I am not being succesfull to find anything related to the study of derivations and locally nilpotent derivations on these sort of ring. What could be a reference for

  1. a description of the derivations on $R$?
  2. a description of the locally nilpotent derivations on $R$?
  3. a description of the automorphism groups of $R$?
  4. any condition for the group of automorphisms of $R$ to be finite/finitely generated?
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    $\begingroup$ What do you call a derivation on a surface? If you mean a tangent vector field, they do not exist ($\neq 0$) on a K3. $\endgroup$
    – abx
    Dec 27 '19 at 17:20
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    $\begingroup$ @abx: "do not exist" means "=0". But the question is, indeed, quite unclear. $\endgroup$
    – Sasha
    Dec 27 '19 at 19:38
  • $\begingroup$ It was really bad written and it was missing an essential part of the data. Sorry for my mistake in not double checking it. I tried to clarify now... $\endgroup$
    – Binai
    Dec 28 '19 at 3:52
  • $\begingroup$ 1. is just given by the Euler sequence. They are just derivations of the polynomial ring (which are obvious) which takes the equation of the hypersurface to a multiple of the same equation. $\endgroup$
    – Mohan
    Dec 29 '19 at 15:57
  • $\begingroup$ @Mohan It is pretty clear that the derivations should have this property related to the polynomial ideal that we are taking the quotient, but I want to know a characterization (if it exists). $\endgroup$
    – Binai
    Dec 31 '19 at 5:14
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I can try to say a little about your first question, on derivations of $R$, in characteristic 0. As Mohan said, the derivations of $R$ are just derivations of the polynomial ring taking the defining equation to itself, but I'd add that it's a bit subtle what the collection of such derivations looks like. It's clear we have derivations of every positive degree $k$, for example by first applying the Euler operator and then multiplying by a polynomial of degree $k$. So, one natural question you can ask is whether there's a derivation of $R$ of negative degree. Note that such a derivation is locally nilpotent (though not all locally nilpotent derivations have negative degree).

The general context for your question is that we're given a ring $R=S(X,L):=\bigoplus H^0(X,L^m)$, which is naturally the section ring of a (smooth) projective variety with a chosen ample line bundle. By a theorem of J.M. Wahl ("A cohomological characterization of $\mathbb P^n$") if $R$ admits a derivation of negative degree, then $X$ must in fact be $\mathbb P^n$. Thus, in your example there will be no derivations of negative degree. (This crucially uses the characteristic 0 hypothesis.)

In fact, one can rule out differential operators of negative degree of any order on your $R$, not just derivations: one can show that the existence of a differential operator of order $m$ and degree $-e$ gives rise to an element of $H^0(X,(\mathrm{Sym}^m T_X)(-e))$, and in turn to an ample subsheaf of $\mathrm{Sym}^m T_X$. By powerful results of Miyaoka (Corollary 8.6 of "Deformations of a Morphism along a Foliation and Applications") this in turn forces $X$ to be uniruled (again, in characteristic 0!), which your K3 surface is not.

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  • $\begingroup$ It is pretty clear that the derivations should have this property related to the polynomial ideal that we are taking the quotient, but I want to know a description (if it exists), i.e, what "the collection of such derivations looks like". $\endgroup$
    – Binai
    Dec 31 '19 at 5:12
  • $\begingroup$ Ah, for sure that characteristic 0 is a crucial step. I would be happy with answers in this case! $\endgroup$
    – Binai
    Dec 31 '19 at 5:16

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