Automorphisms of surfaces in a class of Wehler \(K3\) surfaces with Picard number \(4\). (English) Zbl 1358.14028

Baragar, Arthur

Rocky Mt. J. Math. 46, No. 2, 399-412 (2016).

The paper under review concerns the so-called Wehler \(K3\) surfaces, i.e. smooth surfaces given by a trihomogeneous \((2,2,2)\)-form in \(\mathbb P^1\times\mathbb P^1\times\mathbb P^1\). Generically, such a surface has Picard number three, generated by the fibers of the three obvious projections, each of which generally is an elliptic curve.
The author studies more specifically those Wehler \(K3\) surfaces of Picard number 4 where two of the fibrations attain a section (which features as a component of a reducible fiber of the third fibration). These \(K3\) surfaces are endowed with a set of four canonical involutions: three of them induced from the quadratic form structure over each \(\mathbb P^1\), one coming from either of the elliptic fibrations (with section!) which thus admit a hyperelliptic involution.
Based on the Torelli theorem and lattice theory, the author proves that these four involutions generate the automorphism group of the \(K3\) surface up to finite index. It is also stated that the techniques apply to other \(K3\) surfaces as well, especially those with small Picard number.

Reviewer: Matthias Schütt (Hannover)

MSC:

14J27	Elliptic surfaces, elliptic or Calabi-Yau fibrations
14J28	\(K3\) surfaces and Enriques surfaces
14J50	Automorphisms of surfaces and higher-dimensional varieties

Keywords:

\(K3\) surface; automorphism; ample cone; lattice; isometry; hyperbolic space

References:

[1]	A. Baragar, Rational points on \(K3\) surfaces in \({\mathbf P}^1\times{\mathbf P}^1\times{\mathbf P}^1\) , Math. Ann. 305 (1996), 541-558. · Zbl 0877.14017 · doi:10.1007/BF01444236
[2]	Arthur Baragar, The ample cone for a \(K3\) surface , Canad. J. Math. 63 (2011), 481-499. · Zbl 1219.14049 · doi:10.4153/CJM-2011-006-7
[3]	Arthur Baragar and Ronald van Luijk, \(K3\) surfaces with Picard number three and canonical vector heights , Math. Comp. 76 (2007), 1493-1498 (electronic). · Zbl 1109.14031 · doi:10.1090/S0025-5718-07-01962-X
[4]	Hervé Billard, Propriétés arithmétiques d’une famille de surfaces \(K3\) , Comp. Math. 108 (1997), 247-275. · Zbl 0959.11031 · doi:10.1023/A:1000128300511
[5]	D.R. Morrison, On \(K3\) surfaces with large Picard number , Invent. Math. 75 (1984), 105-121. · Zbl 0509.14034 · doi:10.1007/BF01403093
[6]	I.I. Pjateckiĭ-Šapiro and I.R. Šafarevič, Torelli’s theorem for algebraic surfaces of type \({\mathrm K}3\) , Izv. Akad. Nauk 35 (1971), 530-572.
[7]	Lan Wang, Rational points and canonical heights on \(K3\)-surfaces in \({\mathbf P}^1\times{\mathbf P}^1\times{\mathbf P}^1\) , Contemp. Math. 186 , American Mathematical Society, Providence, RI, 1995. · doi:10.1090/conm/186/02187
[8]	Joachim Wehler, \(K3\)-surfaces with Picard number \(2\) , Arch. Math. (Basel) 50 (1988), 73-82. · Zbl 0602.14038 · doi:10.1007/BF01313498