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Isomorphic quartic $$K3$$ surfaces in the view of Cremona and projective transformations. (English) Zbl 1391.14076
The aim is to study gaps between (1) Cremona equivalence and isomorphism as abstract varieties and (2) Cremona isomorphism and projectively equivalence, for complex smooth quartic surfaces in the projective space.
It is found in Theorem 1.4 that there exists a pair $$(S_1,\, S_2)$$ of $$K3$$ surfaces that is isomorphic as abstract varieties, but not Cremona equivalent, which is shown by an indirect proof. Explicitely, let $$S$$ be a $$K3$$ surfacae with Néron-Severi lattice being $$\langle h_1,\, h_2\rangle_{\mathbb{Z}}$$ with $$(h_i^2)_S = 4$$ for $$i=1,2$$, and $$(h_1.h_2)_S=4l$$, where $$l\geq 5$$, and then $$S_i:=\Phi_{|h_i|}(S)$$ for $$i=1,2$$ would do.
They also construct a pair $$(S_1,\, S_2)$$ of $$K3$$ surfaces in Theorem 1.5, which are Cremona isomorphic but not projective equivalent. In fact, that a surface $$S$$ is a $$K3$$ surface with the Néron-Severi lattice $$\langle h_1,\, h_2\rangle_{\mathbb{Z}}$$ with $$(h_i^2)_S=4$$ for $$i=1,2$$, and $$(h_1.h_2)_S=6$$ is equivalent to that the surface $$S$$ is the intersection of four hypersurfaces of type $$(1,1)$$ in $$\mathbb{P}^3\times\mathbb{P}^3$$. And the pair $$(S_1,\, S_2)$$ with $$S_i:=\Phi_{|h_i|}$$ for $$i=1,2$$ satisfies the desired condition. The key observations are firstly, that the surfaces $$S_1$$ and $$S_2$$ are determinantal surfaces in the sense of Beauville, secondly to study how automorphisms of $$S$$ act on hyperplane sections $$H_1$$ and $$H_2$$, and on $$S_1$$ and $$S_2$$, and lastly, to study how automorphisms of $$S$$ act on the nowhere-vanishing holomorphic form $$\sigma_S$$ of $$S$$.
In Appendix, they give a proof for Proposition 1.7 : the existence of certain curves on smooth quartic surfaces in $$\mathbb{P}^3$$ when it possesses $$\phi\in\text{Bir}(\mathbb{P}^3)\backslash\text{Aut}(\mathbb{P}^3)$$.

##### MSC:
 14J28 $$K3$$ surfaces and Enriques surfaces 14J50 Automorphisms of surfaces and higher-dimensional varieties 14E07 Birational automorphisms, Cremona group and generalizations
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