The Rigidity Theorem of Fano–Segre–Iskovskikh–Manin–Pukhlikov–Corti–Cheltsov–de Fernex–Ein–Mustaţă–Zhuang

Kollár, János

doi:10.1007/978-3-030-18638-8_4

Cited by 11 publications

(7 citation statements)

References 48 publications

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“…Thus it remains to prove thatσ n ≥ 1 n 2n−2 n−1 . This comes from the fact that if (1, 1, · · · , 1) ∈ Q a so that n i=1 a i ≤ 1 and m 1 , · · · , m n ∈ Z satisfy n i=1 m i ≤ n − 1, then at least one cyclic permutation of (m 1 , · · · , m n ) lies in Q a (see [Kol18,Paragraph 57]).…”

Section: Counterexamplementioning

confidence: 99%

Birational superrigidity is not a locally closed property

Zhuang

2020

Sel. Math. New Ser.

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We prove an optimal result on the birational rigidity and K-stability of index 1 hypersurfaces in P n+1 with ordinary singularities when n ≫ 0 and also study the birational superrigidity and K-stability of certain weighted complete intersections. As an application, we show that birational superrigidity is not a locally closed property in moduli. We also prove (in the appendix) that the alpha invariant function is constructible in some families of complete intersections.

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Section: Counterexamplementioning

confidence: 99%

Birational superrigidity is not a locally closed property

Zhuang

2020

Sel. Math. New Ser.

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“…I call it the Noether-Fano-Segre-Iskovskikh-Manin-Pukhlikov-Corti-Cheltsovde Fernex-Ein-Mustaţȃ-Zhuang theorem, although the contributions of Fano, Iskovskikh, and Pukhlikov were the most substantial. See [Kol18] for a detailed survey.…”

Section: Rigidity Of Low Degree Hypersurfacesmentioning

confidence: 99%

Algebraic hypersurfaces

Kollár

2019

Bull. Amer. Math. Soc.

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We give an introduction to the study of algebraic hypersurfaces, focusing on the problem of when two hypersurfaces are isomorphic or close to being isomorphic. Working with hypersurfaces and emphasizing examples makes it possible to discuss these questions without any previous knowledge of algebraic geometry. At the end we formulate the main recent results and state the most important open questions. Algebraic geometry started as the study of plane curves C ⊂ R 2 defined by a polynomial equation and later extended to surfaces and higher dimensional sets defined by systems of polynomial equations. Besides using R n , it is frequently more advantageous to work with C n or with the corresponding projective spaces RP n and CP n. Later it was realized that the theory also works if we replace R or C by other fields, for example the field of rational numbers Q or even finite fields F q. When we try to emphasize that the choice of the field is pretty arbitrary, we use A n to denote affine n-space and P n to denote projective n-space.

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“…It is worth to compare the results on variation of stable birational types in [Shi19] and the present paper with the concept of birational rigidity (see [Kol19] for a recent survey), which allows to prove that for certain classes of smooth projective Fano varieties of Picard rank one, any birational equivalence is an isomorphism. In particular, in these cases the birational types vary as much as possible.…”

Section: Introductionmentioning

confidence: 97%

Variation of stable birational types in positive characteristic

Schreieder¹

2020

Épijournal De Géométrie Algébrique

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Let k be an uncountable algebraically closed field and let Y be a smooth projective k-variety which does not admit a decomposition of the diagonal. We prove that Y is not stably birational to a very general hypersurface of any given degree and dimension. We use this to study the variation of the stable birational types of Fano hypersurfaces over fields of arbitrary characteristic. This had been initiated by Shinder, whose method works in characteristic zero. Comment: 14 pages; final version, published in EPIGA

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The Rigidity Theorem of Fano–Segre–Iskovskikh–Manin–Pukhlikov–Corti–Cheltsov–de Fernex–Ein–Mustaţă–Zhuang

Cited by 11 publications

References 48 publications

Birational superrigidity is not a locally closed property

Birational superrigidity is not a locally closed property

Algebraic hypersurfaces

Variation of stable birational types in positive characteristic

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