A hypersurface is a generalization of an ordinary two-dimensional surface embedded in three-dimensional space to an 
-dimensional surface
embedded in 
-dimensional space. Locally,
a smooth hypersurface can be written as the regular level
set
 |
where the gradient of 
does not vanish, and so it has codimension
one. For instance, the 
-dimensional hypersphere 
in 
corresponds to the equation 
.
Hotchkiss (2026) reported that a very general projective hypersurface of degree at least 5 and positive dimension, over an uncountable algebraically closed field of field characteristic different from 2, has no decomposition of the diagonal. Consequently, it is neither a stably rational variety nor a retract rational variety. Here very general excludes a countable union of proper closed subsets of the parameter space. Hotchkiss (2026) credits ChatGPT 6 Astra with the final construction and most proof arguments. He states that he wrote the details and is responsible for the paper's correctness.
Complex analytic hypersurfaces also include singular zero sets of holomorphic functions. At the origin, a reduced defining germ has multiplicity equal to the least total degree of its nonzero terms. The Zariski multiplicity conjecture concerns whether a homeomorphism of the ambient pair germs preserves that integer.
