Kulikov, Vik. S. 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 \(h^0(X,\mathscr O_X(D+C))=1\). Cited in 3 Documents MSC: 14J29 Surfaces of general type 14C20 Divisors, linear systems, invertible sheaves 13N15 Derivations and commutative rings Keywords:pre-spectral data for commutative subalgebras of rank 1; algebras of differential operators; Godeaux surfaces PDFBibTeX XMLCite \textit{Vik. S. Kulikov}, Sb. Math. 209, No. 8, 1155--1163 (2018; Zbl 1428.14068); translation from Mat. Sb. 209, No. 8, 56--65 (2018) Full Text: DOI References: [1] Barth, W. P.; Hulek, K.; Peters, C. A. M.; Ven, A. Van de, Ergeb. Math. Grenzgeb. (3), 4, (2004), Springer-Verlag: Springer-Verlag, Berlin-Heidelberg · Zbl 1036.14016 [2] Burban, I.; Zheglov, A., Cohen-Macaulay modules over the algebra of planar quasi- invariants and Calogero-Moser systems · Zbl 1453.14011 [3] Bourbaki, N., Actualités Sci. Indust., 1337, (1968), Hermann & Cie: Hermann & Cie, Paris · Zbl 0186.33001 [4] Zheglov, A. B.; Osipov, D. V., On some questions related to the Krichever correspondence, Mat. Zametki, 81, 4, 528-539, (2007) · Zbl 1134.14023 [5] Krichever, I. M., Commutative rings of ordinary linear differential operators, Funktsional. Anal. i Prilozhen., 12, 3, 20-31, (1978) · Zbl 0408.34008 [6] Kulikov, Vik. S., Old and new examples of surfaces of general type with, Izv. Ross. Akad. Nauk Ser. Mat., 68, 5, 123-170, (2004) · Zbl 1073.14055 [7] Kulikov, Vik. S., Dualizing coverings of the plane, Izv. Ross. Akad. Nauk Ser. Mat., 79, 5, 163-192, (2015) · Zbl 1355.14011 [8] Kulikov, Vik. S., Plane rational quartics and K3 surfaces, Proc. Steklov Inst. Math., 294, 105-140, (2016) · Zbl 1356.14028 [9] Kulikov, Vik. S.; Shustin, E. I., On, Proc. Steklov Inst. Math., 298, 144-164, (2017) · Zbl 1396.14040 [10] Kulikov, Vik. S.; Kharlamov, V. M., On real structures on rigid surfaces, Izv. Ross. Akad. Nauk Ser. Mat., 66, 1, 133-152, (2002) · Zbl 1055.14060 [11] Kulikov, Vik. S.; Kharlamov, V. M., Surfaces with, Izv. Ross. Akad. Nauk Ser. Mat., 70, 4, 135-174, (2006) · Zbl 1153.14027 [12] Kurke, H.; Osipov, D.; Zheglov, A., Commuting differential operators and higher- dimensional algebraic varieties, Selecta Math. (N.S.), 20, 4, 1159-1195, (2014) · Zbl 1306.37077 [13] Zheglov, A. B.; Kurke, H., Geometric properties of commutative subalgebras of partial differential operators, Mat. Sb., 206, 5, 61-106, (2015) · Zbl 1329.13043 [14] Manetti, M., On the moduli space of diffeomorphic algebraic surfaces, Invent. Math., 143, 1, 29-76, (2001) · Zbl 1060.14520 [15] Osipov, D. V., Krichever correspondence for algebraic varieties, Izv. Ross. Akad. Nauk Ser. Mat., 65, 5, 91-128, (2001) · Zbl 1068.14053 [16] Parshin, A. N., On a ring of formal pseudodifferential operators, Proc. Steklov Inst. Math., 224, 291-305, (1999) · Zbl 1008.37042 [17] Parshin, A. N., The Krichever correspondence for algebraic varieties, Funktsional. Anal. i Prilozhen., 35, 1, 88-90, (2001) · Zbl 1078.14525 [18] Przyjalkowski, V. V.; Shramov, C. A., On weak Landau-Ginzburg models for complete intersections in Grassmannians, Uspekhi Mat. Nauk, 69, 6-420, 181-182, (2014) · Zbl 1319.14047 [19] Przyjalkowski, V.; Shramov, C., On Hodge numbers of complete intersections and Landau-Ginzburg models, Int. Math. Res. Not., 2015, 21, 11302-11332, (2015) · Zbl 1343.14038 [20] Przyjalkowski, V. V.; Shramov, C. A., Laurent phenomenon for Landau-Ginzburg models of complete intersections in Grassmannians, Proc. Steklov Inst. Math., 290, 102-113, (2015) · Zbl 1427.14082 [21] Przyjalkowski, V. V., Calabi-Yau compactifications of toric Landau-Ginzburg models for smooth Fano threefolds, Mat. Sb., 208, 7, 84-108, (2017) · Zbl 1386.14055 [22] Przyjalkowski, V.; Shramov, C., Laurent phenomenon for Landau-Ginzburg models of complete intersections in Grassmannians of planes, Bull. Korean Math. Soc., 54, 5, 1527-1575, (2017) · Zbl 1401.14177 [23] Serre, J.-P., Cours d’arithmétique, (1970), Presses Universitaires de France: Presses Universitaires de France, Paris · Zbl 0225.12002 [24] Zheglov, A. B., On rings of commuting partial differential operators, Algebra i Analiz, 25, 5, 86-145, (2013) · Zbl 1325.13025 [25] Zheglov, A. B., Surprising example of nonrational smooth spectral surfaces, Mat. Sb., 209, 8, 25-55, (2018) · Zbl 1408.13069 [26] Zheglov, A. B., Two dimensional KP systems and their solvability This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.