ICM 2026 Invited Section Lecture - Stefan Schreieder

Value of the Information & Strength of the Argument

The talk presents high-value mathematical content: it outlines a coherent research program, connects classical results with modern techniques, and clearly explains the logical chain from definitions to theorems. The argumentation is rigorous, with each result placed in context and the methods (algebraic cycles, specialization, vanishing theorems) carefully motivated. The speaker’s expertise allows him to convey both the breadth of the field and the technical subtleties without oversimplifying, making the presentation valuable for specialists.

Scientific Rigor, Source Quality, Title Accuracy

The talk exhibits strong scientific rigor: the speaker cites specific works (e.g., Euler, Clemens-Griffiths, Kollár, Totaro) and his own joint papers, and the logical progression from classical to modern results is clear. While no explicit bibliographic links are provided in the video description, the verbal references are accurate and can be traced in the literature. The title is perfectly aligned with the content, as the entire talk addresses the rationality of hypersurfaces. The speaker also highlights open cases, demonstrating intellectual honesty and precision.

The title accurately reflects the content, as the lecture focuses specifically on the rationality of hypersurfaces and related notions.

8/10

The talk is given by a leading researcher in algebraic geometry, presenting original results from recent joint papers. The arguments are logically rigorous, though the content is highly specialized and assumes advanced background.

Markers derived by PSI from the transcript: the creator did not define chapters.

• 02:00Introduction of the classical problem: Pythagorean triples and rational points on the circle via stereographic projection.
• 05:00Definition of rational varieties and the historical example of Euler's cubic surface (1761).
• 08:00Introduction of retract rationality and its equivalent formulations, including the role of rational sections.
• 12:00Discussion of known results for cubic hypersurfaces: curves, surfaces, threefolds, and fourfolds.
• 18:00Presentation of new results: the very general cubic threefold is not retract rational (joint work with Engel and de Gaay Fortman).
• 24:00Explanation of the logarithmic bound for non-retract-rational hypersurfaces and later improvements.
• 30:00Overview of proof techniques: decomposition of the diagonal, specialization, and unramified cohomology.

• Euler's paper on cubic surfaces (1761) — The speaker cites Euler's 1761 paper where he parametrized the rational solutions of the smooth cubic surface.
• Clemens-Griffiths (1972) - 'The intermediate Jacobian of the cubic threefold' — Quoted as the first proof that smooth cubic threefolds are not rational.
• Kollár (1995) - 'Nonrational hypersurfaces' — Referenced for showing that very general hypersurfaces of sufficiently high degree relative to dimension are not rational.
• Totaro (2016) - work on retract rationality — Mentioned as improving Kollár's bound to establish non-retract-rationality under similar degree conditions.
• Hassett, Perutka, Tschinkel - on irrationality of certain fourfolds — Used in reduction arguments for lower-dimensional cases.
• Engel, de Gaay Fortman, Schreieder - joint work on retract rationality of cubic threefolds — The speaker's own result proving that the very general cubic threefold is not retract rational.