Algebraic cycles on hypersurfaces

• Applicant: Dr. Matthias Paulsen
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 558829442

In algebraic geometry, hypersurfaces are one of the simplest and most important examples of algebraic varieties. An algebraic variety is the set of solutions of a system of polynomial equations. In the case of hypersurfaces, this system of equations consists only of a single equation. The geometric properties of such a hypersurface depend heavily on how large the degree of its defining equation is compared to its dimension, that is, the number of variables. In particular, an interesting geometric property is which other varieties are contained in a given one. Linear combinations of such sub-varieties are also called algebraic cycles. In the case of hypersurfaces, a generalization of a conjecture by Griffiths and Harris from 1985 predicts that the degree of all algebraic cycles having positive dimension on a very general hypersurface with sufficiently large degree is already divisible by the degree of the hypersurface. Recently, based on older results by Kollár, crucial progress on this conjecture was obtained by the author of the project, through completely proving the conjecture for the first time for certain degrees. In each dimension, there exist infinitely many degrees where this was possible. However, the conjecture remains open for infinitely many degrees as well. One of the objectives of this project is to achieve further progress on the conjecture by Griffiths and Harris, through extending the set of degrees where the conjecture can be proven. Another goal concerns the failure of the integral Hodge conjecture on 3-dimensional hypersurfaces of low degree. This problem is also deeply linked to the possible degrees of algebraic cycles on a hypersurface. Finally, 0-dimensional cycles, i.e. linear combination of points, on hypersurfaces are investigated during this project and their possible degrees are studied. In particular, the question whether the existence of a 0-cycle of degree 1 on a cubic hypersurface implies the existence of a rational point is considered. This question is only relevant over fields which are not algebraically closed, while the first two problems are studied mostly over the field of complex numbers.

• DFG Programme: WBP Fellowship
• International Connection: France
• Host: Dr. Olivier Benoist