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On divisors of small canonical degree on Godeaux surfaces. (English. Russian original) Zbl 1428.14068

Sb. Math. 209, No. 8, 1155-1163 (2018); translation from Mat. Sb. 209, No. 8, 56-65 (2018).
Summary: Pre-spectral data \((X,C,D)\) coding the rank-1 commutative subalgebras of a certain completion \(\widehat D\) of the algebra of differential operators \(D=k[[x_1,x_2]][\partial_1,\partial_2]\), where \(k\) is an algebraically closed field of characteristic 0, are shown to exist. Here \(X\) is a Godeaux surface, \(C\) is an effective ample divisor represented by a smooth curve, \(h^0(X,\mathscr O_X(C))=1\) and \(D\) is a divisor on \(X\) satisfying the conditions \((D, C)_X=g(C)-1\), \(h^i(X,\mathscr O_X(D))=0\) for \(i=0,1,2\) and \(h^0(X,\mathscr O_X(D+C))=1\).

MSC:

14J29 Surfaces of general type
14C20 Divisors, linear systems, invertible sheaves
13N15 Derivations and commutative rings
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