Surprising examples of nonrational smooth spectral surfaces
A. B. Zheglov

References
1.	L. Bădescu, Projective geometry and formal geometry, IMPAN Monogr. Mat. (N. S.), 65, Birkhäuser Verlag, Basel, 2004, xiv+209 pp.
2.	Yu. Berest, P. Etingof, V. Ginzburg, “Cherednik algebras and differential operators on quasi-invariants”, Duke Math. J., 118:2 (2003), 279–337
3.	Yu. Berest, A. Kasman, “$\mathscr D$-modules and Darboux transformations”, Lett. Math. Phys., 43:3 (1998), 279–294
4.	I. Burban, A. Zheglov, Cohen–Macaulay modules over the algebra of planar quasi-invariants and Calogero–Moser systems, arXiv: 1703.01762
5.	N. Bourbaki, Éléments de mathématique. Algèbre commutative, v. 27, 28, 30, 31, Actualités Sci. Indust., 1290, 1293, 1308, 1314, Hermann, Paris, 1961–1965, 187 pp., 183 pp., 207 pp., iii+146 pp.
6.	O. Chalykh, “Algebro-geometric Schrödinger operators in many dimensions”, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 366:1867 (2008), 947–971
7.	O. A. Chalykh, A. P. Veselov, “Commutative rings of partial differential operators and Lie algebras”, Comm. Math. Phys., 126:3 (1990), 597–611
8.	A. P. Veselov, K. L. Styrkas, O. A. Chalykh, “Algebraic integrability for the Schrödinger equation and finite reflection groups”, Theoret. and Math. Phys., 94:2 (1993), 182–197
9.	O. Chalykh, M. Feigin, A. Veselov, “New integrable generalizations of Calogero–Moser quantum problem”, J. Math. Phys., 39:2 (1998), 695–703
10.	B. A. Dubrovin, “Matrix finite-zone operators”, J. Soviet Math., 28:1 (1985), 20–50
11.	B. A. Dubrovin, V. B. Matveev, S. P. Novikov, “Non-linear equations of Korteweg–de Vries type, finite-zone linear operators, and Abelian varieties”, Russian Math. Surveys, 31:1 (1976), 59–146
12.	B. A. Dubrovin, I. M. Krichever, S. P. Novikov, “Integrable systems. I”, Dynamical systems IV, Encyclopaedia Math. Sci., 4, Springer, Berlin, 1990, 173–280
13.	M. Feigin, D. Johnston, “A class of Baker–Akhiezer arrangements”, Comm. Math. Phys., 328:3 (2014), 1117–1157
14.	P. G. Grinevich, “Vector rank of commuting matrix differential operators. Proof of S. P. Novikov's criterion”, Math. USSR-Izv., 28:3 (1987), 445–465
15.	A. Grothendieck, “Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes”, Inst. Hautes Études Sci. Publ. Math., 8 (1961), 5–222
16.	A. Grothendieck, “Éléments de géométrie algèbrique. IV. Étude locale des schémas et des morphismes de schémas. III”, Inst. Hautes Études Sci. Publ. Math., 28 (1966), 5–255
17.	R. Hartshorne, Algebraic geometry, Grad. Texts in Math., 52, Springer-Verlag, New York–Heidelberg, 1977, xvi+496 pp.
18.	G. J. Heckman, “A remark on the Dunkl differential-difference operators”, Harmonic analysis on reductive groups (Brunswick, ME, 1989), Progr. Math., 101, Birkhäuser Boston, Boston, MA, 1991, 181–191
19.	G. J. Heckman, E. M. Opdam, “Root systems and hypergeometric functions. I”, Compositio Math., 64:3 (1987), 329–352
20.	I. M. Krichever, “Methods of algebraic geometry in the theory of non-linear equations”, Russian Math. Surveys, 32:6 (1977), 185–213
21.	I. M. Krichever, “Commutative rings of ordinary linear differential operators”, Funct. Anal. Appl., 12:3 (1978), 175–185
22.	I. M. Krichever, S. P. Novikov, “Two-dimensionalized Toda lattice, commuting difference operators, and holomorphic bundles”, Russian Math. Surveys, 58:3 (2003), 473–510
23.	Vik. S. Kulikov, “On divisors of small canonical degree on the Godaux surfaces”, Sb. Math., 209:8 (2018)
24.	H. Kurke, D. Osipov, A. Zheglov, “Commuting differential operators and higher-dimensional algebraic varieties”, Selecta Math. (N. S.), 20:4 (2014), 1159–1195
25.	A. B. Zheglov, H. Kurke, “Geometric properties of commutative subalgebras of partial differential operators”, Sb. Math., 206:5 (2015), 676–717
26.	G. S. Mauleshova, A. E. Mironov, “One-point commuting difference operators of rank 1”, Dokl. Math., 93:1 (2016), 62–64      ; arXiv: 1507.00527
27.	A. E. Mironov, “Commutative rings of differential operators corresponding to multidimensional algebraic varieties”, Siberian Math. J., 43:5 (2002), 888–898
28.	K. Cho, A. Mironov, A. Nakayashiki, “Baker–Akhiezer modules on the intersections of shifted theta divisors”, Publ. Res. Inst. Math. Sci., 47:2 (2011), 553–567
29.	M. Mulase, “Category of vector bundles on algebraic curves and infinite dimensional Grassmanians”, Internat. J. Math., 1:3 (1990), 293–342
30.	M. Mulase, “Algebraic theory of the KP equations”, Perspectives in mathematical physics, Conf. Proc. Lecture Notes Math. Phys., III, Int. Press, Cambridge, MA, 1994, 151–217
31.	A. Nakayashiki, “Commuting partial differential operators and vector bundles over Abelian varieties”, Amer. J. Math., 116:1 (1994), 65–100
32.	A. Nakayashiki, “Structure of Baker–Akhiezer modules of principally polarized Abelian varieties, commuting partial differential operators and associated integrable systems”, Duke Math. J., 62:2 (1991), 315–358
33.	D. V. Osipov, “The Krichever correspondence for algebraic varieties”, Izv. Math., 65:5 (2001), 941–975
34.	S. O. Gorchinskiy, D. V. Osipov, “Continuous homomorphisms between algebras of iterated Laurent series over a ring”, Proc. Steklov Inst. Math., 294 (2016), 47–66
35.	S. O. Gorchinskiy, D. V. Osipov, “Higher-dimensional Contou-Carrère symbol and continuous automorphisms”, Funct. Anal. Appl., 50:4 (2016), 268–280
36.	A. B. Zheglov, D. V. Osipov, “On some questions related to the Krichever correspondence”, Math. Notes, 81:4 (2007), 467–476
37.	M. A. Olshanetsky, A. M. Perelomov, “Quantum integrable systems related to Lie algebras”, Phys. Rep., 94:6 (1983), 313–404
38.	A. N. Parshin, “The Krichever correspondence for algebraic surfaces”, Funct. Anal. Appl., 35:1 (2001), 74–76
39.	A. N. Parshin, “Integrable systems and local fields”, Comm. Algebra, 29:9 (2001), 4157–4181
40.	E. Previato, “Multivariable Burchnall–Chaundy theory”, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 366:1867 (2008), 1155–1177
41.	V. Przyjalkowski, C. Shramov, “On Hodge numbers of complete intersections and Landau–Ginzburg models”, Int. Math. Res. Not. IMRN, 2015:21 (2015), 11302–11332
42.	V. V. Przyjalkowski, “Calabi–Yau compactifications of toric Landau–Ginzburg models for smooth Fano threefolds”, Sb. Math., 208:7 (2017), 992–1013
43.	M. Rothstein, “Dynamics of the Krichever construction in several variables”, J. Reine Angew. Math., 2004:572 (2004), 111–138
44.	A. B. Zheglov, “On rings of commuting partial differential operators”, St. Petersburg Math. J., 25:5 (2014), 775–814
45.	A. B. Zheglov, Puchki bez krucheniya na mnogoobraziyakh i integriruemye sistemy, Diss. … dokt. fiz.-matem. nauk, Matem. in-t im. V. A. Steklova RAN, M., 2016, 201 pp. http://www.mi.ras.ru/dis/ref16/zheglov/dis.pdf