Deformations of some algebraic surfaces with \(q=0\) and \(p_ g=1\)
Let \(M^ a\) be an affine algebraic surface in \({\mathbb{C}}^ 3\) defined by \(h(w)=1+w^ 8_ 1+w^ 2_ 1W^ 4_ 3+W^ 3_ 1W^ 2_ 2=0\). Then \(M^ a\) admits a smooth compactification M, embedded in a 3- dimensional toric variety W. There are 27 additional integral points \(A_ 1\) on the simplex \(\Delta\) spanned by (8,0,0), (2,0,4), (3,2,0), and they determine a family \(h(w)+\sum t_ iw^{A_ i} =0\) of deformations of \(M_ 0\). They can be compactified to a family \(M_ t\) of deformations of \(M=M_ 0\) in W. The author studies how this is related to all the small deformations of M: The family generates a subspace of codimension 2 in the infinitesimal deformation space \(H^ 1(M,\Theta)\). The surface M itself is an elliptic surface with \(p_ g=1\), \(q=0\) and \(\pi_ 1(M)={\mathbb{Z}}/2{\mathbb{Z}}\).
scientific article; zbMATH DE number 4079589