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The isomorphism problem of projective schemes and related algorithmic problems

Takehiko Yasuda

Department of Mathematics, Graduate School of Science, Osaka University Toyonaka, Osaka 560-0043, Japan

E-mail Address: yasuda.takehiko.sci@osaka-u.ac.jp

https://doi.org/10.1142/S021819672350039XCited by:0 (Source: Crossref)

Abstract

We discuss the isomorphism problem of projective schemes; given two projective schemes, can we algorithmically decide whether they are isomorphic? We give affirmative answers in the case of one-dimensional projective schemes, the case of smooth irreducible varieties with a big canonical sheaf or a big anti-canonical sheaf, and the case of K3 surfaces with a finite automorphism group. As related algorithmic problems, we also discuss decidability of positivity properties of invertible sheaves, and approximation of the nef cone and the pseudo-effective cone.

Communicated by J. McCullough

Keywords:

AMSC: 14Q15, 14Q10, 14Q05

References

1. M. H. Baker, E. González-Jiménez, J. González and B. Poonen , Finiteness results for modular curves of genus at least 2, Am. J. Math. 127(6) (2005) 1325–1387. Crossref, Web of Science, Google Scholar
2. W. Barth, K. Hulek, C. Peters and A. v. d. Ven , Compact Complex Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge/ A Series of Modern Surveys in Mathematics, 2nd edn. (Springer-Verlag, Berlin Heidelberg, 2004). Crossref, Google Scholar
3. D. Bayer, The division algorithm and the hilbert scheme, Ph.D. Thesis, Harvard University (1982). Google Scholar
4. C. Birkar , On existence of log minimal models II, J. für Reine Angew. Math. 2011 (2011) 99–113. Crossref, Web of Science, Google Scholar
5. E. Bombieri , Canonical models of surfaces of general type, Inst. Hautes Études Sci. Publ. Math. 42 (1973) 171–219. Crossref, Google Scholar
6. C. J. Bott, S. H. Hassanzadeh, K. Schwede and D. Smolkin, RationalMaps, a package for Macaulay2 (2019), arXiv:1908.04337. Google Scholar
7. S. Boucksom, J.-P. Demailly, M. Păun and T. Peternell , The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension, J. Algebr. Geom. 22(2) (2013) 201–248. Crossref, Web of Science, Google Scholar
8. J. A. Chen and M. Chen , Explicit birational geometry of threefolds of general type, I, Ann. Sci. Ec. Norm. Super. 43(3) (2010) 365–394. Crossref, Web of Science, Google Scholar
9. A. L. Chistov , Algorithm of polynomial complexity for factoring polynomials and finding the components of varieties in subexponential time, J. Sov. Math. 34 (1986) 1838–1882. Crossref, Google Scholar
10. T.-C. Dinh and K. Oguiso , A surface with discrete and nonfinitely generated automorphism group, Duke Math. J. 168 (2019) 941–966. Crossref, Web of Science, Google Scholar
11. A. V. Doria, S. H. Hassanzadeh and A. Simis , A characteristic-free criterion of birationality, Adv. Math. 230 (2012) 390–413. Crossref, Web of Science, Google Scholar
12. D. Eisenbud , Commutative Algebra: With a View Toward Algebraic Geometry, Graduate Texts in Mathematics (Springer-Verlag, New York, 1995). Crossref, Google Scholar
13. D. Eisenbud, C. Huneke and W. Vasconcelos , Direct methods for primary decomposition, Invent. Math. 110 (1992) 207–235. Crossref, Web of Science, Google Scholar
14. D. Eisenbud and F.-O. Schreyer , Relative Beilinson monad and direct image for families of coherent sheaves, Trans. Am. Math. Soc. 360 (2008) 5367–5396. Crossref, Web of Science, Google Scholar
15. D. R. Grayson and M. E. Stillman, Macaulay2, a software system for research in algebraic geometry, Available at http://www.math.uiuc.edu/Macaulay2/. Google Scholar
16. G.-M. Greuel and G. Pfister , A Singular Introduction to Commutative Algebra, 2nd, extended edn. (Springer, Berlin, New York, 2008). Google Scholar
17. C. D. Hacon and J. McKernan , Boundedness of pluricanonical maps of varieties of general type, Invent. Math. 166 (2006) 1–25. Crossref, Web of Science, Google Scholar
18. R. Hartshorne , Algebraic Geometry, Graduate Texts in Mathematics (Springer-Verlag, New York, 1977). Crossref, Google Scholar
19. K. Hashizume , On the Non-vanishing conjecture and existence of log minimal models, Publ. Res. Inst. Math. Sci. 54 (2018) 89–104. Crossref, Web of Science, Google Scholar
20. A. Iarrobino and V. Kanev , Sums of powers of linear forms, and Gorenstein algebras, in Power Sums, Gorenstein Algebras, and Determinantal Loci, A. Iarrobino and V. Kanev (eds.), Lecture Notes in Mathematics (Springer, Berlin, Heidelberg, 1999), pp. 57–72. Crossref, Google Scholar
21. A. Y. Kanel’-Belov and A. A. Chilikov , On the algorithmic undecidability of the embeddability problem for algebraic varieties over a field of characteristic zero, Math. Notes 106 (2019) 299–302. Crossref, Web of Science, Google Scholar
22. S. L. Kleiman , Toward a numerical theory of ampleness, Ann. Math. 84(3) (1966) 293–344. Crossref, Web of Science, Google Scholar
23. J. Kollár , Pell surfaces, Acta Math. Hung. 160 (2020) 478–518. Crossref, Web of Science, Google Scholar
24. J. Kollár , Rational Curves on Algebraic Varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge/ A Series of Modern Surveys in Mathematics (Springer-Verlag, Berlin Heidelberg, 1996). Crossref, Google Scholar
25. S. Kondō , The maximum order of finite groups of automorphisms of K3 surfaces, Am. J. Math. 121(6) (1999) 1245–1252. Crossref, Web of Science, Google Scholar
26. S. J. Kovács , The cone of curves of a K3 surface, Math. Ann. 300 (1994) 681–691. Crossref, Web of Science, Google Scholar
27. R. K. Lazarsfeld , Positivity in Algebraic Geometry I: Classical Setting: Line Bundles and Linear Series, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge/ A Series of Modern Surveys in Mathematics, Positivity in Algebraic Geometry (Springer-Verlag, Berlin Heidelberg, 2004). Google Scholar
28. J. Lesieutre , A projective variety with discrete, non-finitely generated automorphism group, Invent. Math. 212 (2018) 189–211. Crossref, Web of Science, Google Scholar
29. H. Matsumura , Commutative Ring Theory, Cambridge Studies in Advanced Mathematics, Vol. 8, 2nd edn. (Cambridge University Press, Cambridge, 1989). Google Scholar
30. R. A. Nicolaides , On a class of finite elements generated by Lagrange interpolation, SIAM J. Numer. Anal. 9(3) (1972) 435–445. Crossref, Web of Science, Google Scholar
31. B. Poonen , Automorphisms mapping a point into a subvariety, J. Algebr. Geom. 20(4) (2011) 785–794. Crossref, Web of Science, Google Scholar
32. B. Poonen , Undecidable problems: A sampler, in Interpreting Gödel: Critical Essays (Cambridge University Press, 2014), pp. 211–241. Crossref, Google Scholar
33. B. Poonen, D. Testa and R. v. Luijk , Computing Néron–Severi groups and cycle class groups, Compos. Math. 151 (2015) 713–734. Crossref, Web of Science, Google Scholar
34. K. Schwede and Z. Yang , Divisor package for Macaulay2, J. Softw. Algebra Geom. 8 (2018) 87–94. Crossref, Google Scholar
35. A. Simis , Cremona transformations and some related algebras, J. Algebra 280 (2004) 162–179. Crossref, Web of Science, Google Scholar
36. C. Simpson , Algebraic cycles from a computational point of view, Theor. Comput. Sci. 392 (2008) 128–140. Crossref, Web of Science, Google Scholar
37. Y.-T. Siu , An effective Matsusaka big theorem, Ann. Inst. Fourier 43(5) (1993) 1387–1405. Crossref, Web of Science, Google Scholar
38. G. G. Smith , Computing Global Extension Modules. J. Symbol. Comput. 29(4) (2000) 729–746. https://doi.org/10.1006/jsco.1999.0399 Crossref, Web of Science, Google Scholar
39. T. Stacks Project Authors, The Stacks Project (2021), https://stacks.math.columbia.edu. Google Scholar
40. H. Sterk , Finiteness results for algebraic K3 surfaces, Math. Z. 189 (1985) 507–513. Crossref, Web of Science, Google Scholar
41. M. Stillman, Computing with sheaves and sheaf cohomology in algebraic geometry: Preliminary version, http://www.fields.utoronto.ca/programs/scientific/06-07/comalgebra/stillman_lecturenotes-m2.pdf. Google Scholar
42. S. Takayama , Pluricanonical systems on algebraic varieties of general type, Invent. Math. 165 (2006) 551. Crossref, Web of Science, Google Scholar
43. T. T. Truong, Bounded birationality and isomorphism problems are computable (2018), arXiv:1801.00901. Google Scholar
44. H. Tsuji , Pluricanonical systems of projective varieties of general type I, Osaka J. Math. 43 (2006) 967–995. Web of Science, Google Scholar
45. R. Vakil, Foundations of algebraic geometry (2015), http://math.stanford.edu/~Vakil/216blog/FOAGapr2915public.pdf. Google Scholar
46. W. Vasconcelos , Computational Methods in Commutative Algebra and Algebraic Geometry, Algorithms and Computation in Mathematics (Springer-Verlag, Berlin Heidelberg, 1998). Crossref, Google Scholar