[Paper Review] Adjunction beyond thresholds and birationally rigid hypersurfaces

Tommaso de Fernex|arXiv (Cornell University)|Apr 10, 2006

Algebraic Geometry and Number Theory18 references2 citations

TL;DR

This paper proves Pukhlikov's conjecture that all smooth hypersurfaces of degree N in projective space P^N are birationally superrigid for N ≥ 4, extending the classical Iskovskikh-Manin result for N = 4. The key advance is a new adjunction formula for singularities of pairs under restriction, generalizing the connectedness principle, and relying on arc space techniques for log-discrepancies from Ein, Lazarsfeld, and Mustaţă.

ABSTRACT

Abstract. We give an affirmative answer to a conjecture of Pukhlikov, proving that for N ≥ 4, all smooth hypersurfaces of degree N in P N are birationally superrigid, the case N = 4 of this result being the celebrated theorem of Iskovskikh and Manin that started this whole direction of research. The main new ingredient to obtain the complete result is an adjunction formula for singularities of pairs under restriction that, under suitable conditions, generalizes the well-known formula for hyperplane sections derived from the connectedness principle of Shokurov and Kollár. The proof uses in an essential way a result on log-discrepancies via arc spaces due to Ein, Lazarsfeld and Mustat¸ǎ.

Motivation & Objective

• To resolve Pukhlikov's conjecture on the birational superrigidity of smooth hypersurfaces of degree N in P^N for N ≥ 4.
• To establish a generalization of the connectedness principle for adjunction of singularities under restriction to hypersurfaces.
• To extend the framework of birational rigidity beyond known threshold cases by developing a new adjunction formula for pairs with singularities.
• To apply arc space techniques for log-discrepancies to control singularities in the context of birational geometry.

Proposed method

• Develop a new adjunction formula for singularities of pairs under restriction to hypersurfaces, generalizing the classical hyperplane section formula.
• Use the connectedness principle of Shokurov and Kollár as a foundational case for the generalized adjunction formula.
• Apply results on log-discrepancies via arc spaces due to Ein, Lazarsfeld, and Mustaţă to analyze singularities in the context of birational rigidity.
• Establish that the generalized adjunction formula holds under suitable conditions on the pair and the restriction divisor.
• Combine the new adjunction formula with existing machinery in birational geometry to prove superrigidity for all smooth hypersurfaces of degree N in P^N for N ≥ 4.
• Verify that the conditions of the adjunction formula are satisfied in the case of smooth hypersurfaces of degree N in P^N, ensuring the absence of birational maps to other Fano varieties.

Experimental results

Research questions

• RQ1 Does every smooth hypersurface of degree N in P^N for N ≥ 4 admit a birational map to a different Fano variety of the same dimension and Picard number?
• RQ2 Can the classical adjunction formula for hyperplane sections be generalized to pairs with singularities under restriction to hypersurfaces?
• RQ3 What conditions ensure that the log-canonical threshold of a pair remains preserved under restriction to a hypersurface?
• RQ4 How do arc space techniques for log-discrepancies contribute to proving birational rigidity in higher-dimensional hypersurfaces?
• RQ5 Is there a uniform adjunction principle that extends the connectedness principle to singular pairs in higher codimension?

Key findings

• All smooth hypersurfaces of degree N in P^N are birationally superrigid for N ≥ 4, confirming Pukhlikov's conjecture.
• A new adjunction formula for singularities of pairs under restriction is established, generalizing the connectedness principle of Shokurov and Kollár.
• The adjunction formula holds under suitable conditions that are satisfied in the case of smooth hypersurfaces of degree N in P^N.
• The proof relies crucially on arc space methods for log-discrepancies developed by Ein, Lazarsfeld, and Mustaţǎ.
• The result extends the classical Iskovskikh-Manin theorem for quartic threefolds to all dimensions N ≥ 4.
• The framework provides a systematic approach to proving birational rigidity beyond the known threshold cases.