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    Scientists / Open questions

    Alexander Grothendieck

    Algebraic geometry; category theory

    1966 Fields Medal

    14 questions

    1. The mixed-characteristic lifting problem for smooth projective varieties (Grothendieck, SGA 1, Exposé III)

    2. The unfinished specialization theory of the fundamental group after SGA 1, Exposé X: higher étale homotopy exact sequences in positive characteristic, purely algebraic presentation of curve fundamental groups, and the continuous component of the full fundamental group

    3. Characterizing Grothendieck rigidity: which finitely generated residually finite groups are left or right Grothendieck rigid

    4. Geometrically connected formal fibres and descent of finite étale coverings for general noetherian henselian local rings (EGA IV, Remarques 18.9.6 (i)–(ii))

    5. Ramified coverings of formal schemes beyond the normality, normal-crossings, and tameness hypotheses of Grothendieck–Murre

    6. Connected non-irreducible finite etale covers of integral schemes with a non-geometrically-unibranch point (SGA 1, Expose I, §11)

    7. Local constructibility of the openness and universal-openness loci of a morphism (EGA IV_3, Remarques 14.1.3(iii) and 14.3.9(ii))

    8. Grothendieck–Murre tame fundamental group of a formal neighbourhood: kernel and extension class for intersecting boundary components (LNM 208, Theorem 7.3.1 without disjointness)

    9. Omitting the direct-summand requirement in axiom (G3) of Grothendieck's Galois categories (SGA 1, Exposé V, Remarques 4.2)

    10. Grothendieck's conjecture on resolution of singularities for reduced quasi-excellent noetherian schemes (EGA IV₂, Remarque 7.9.6)

    11. Characterizing the Fréchet spaces EE for which Grothendieck's germ space P(Ω1,E)P(\Omega_1,E) is complete, semi-reflexive, Montel, or distinguished

    12. Intrinsic criteria for effectivity of descent data in faithfully flat (fpqc) descent of schemes (SGA 1, Exposé VIII, §7)

    13. Grothendieck's 1966 algebraic de Rham questions: the open core — algebraic independence of periods and quasi-periods, Gauss–Manin horizontality of algebraic cycle classes, and E_2-finiteness of the Cousin spectral sequence

    14. Does L⊗n\mathscr{L}^{\otimes n} very ample imply L⊗(n+1)\mathscr{L}^{\otimes (n+1)} very ample? (EGA II, Remark 4.5.12)

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