# K3 Surfaces : Derivations and automorphisms

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?
• 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.
– abx
Dec 27 '19 at 17:20
• @abx: "do not exist" means "=0". But the question is, indeed, quite unclear. Dec 27 '19 at 19:38
• 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... Dec 28 '19 at 3:52
• 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. Dec 29 '19 at 15:57
• @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). Dec 31 '19 at 5:14

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.