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Borcherds’ method for Enriques surfaces. (English) Zbl 1487.14090

R. Borcherds’ method [J. Algebra 111, 133–153 (1987; Zbl 0628.20003); Int. Math. Res. Not. 1998, No. 19, 1011–1031 (1998; Zbl 0935.20027)] is a procedure to calculate the automorphism group \(\mathrm{Aut}(X)\) of a \(K3\) surface \(X\) by embedding \(S_X\) primitively into \(L_{26}\) and by applying J. H. Conway’s result [J. Algebra 80, 159–163 (1983; Zbl 0508.20023)] on the orthogonal group \(\mathrm{O}(L_{26})\) of \(L_{26}\).
Let \(Y\) be an Enriques surface in characteristic \(\not= 2\) with the universal covering \(\pi:X\rightarrow Y\) and let \(S_Y\) (resp. \(S_X\)) be the lattice of numerical equivalence classes of divisors on \(Y\) (resp. \(X\)). Then we have a primitive embedding \(\pi^*:S_Y(2)\hookrightarrow S_X\). Note that \(S_Y\) is isomorphic to \(L_{10}\).

MSC:

14J50 Automorphisms of surfaces and higher-dimensional varieties
14J28 \(K3\) surfaces and Enriques surfaces

Software:

PARI/GP; Magma; GAP; SageMath
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Full Text: DOI arXiv

References:

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