Algebraic hypersurfaces

Abstract

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.

Trisecant Flops, their associated K3 surfaces and the rationality of some Fano fourfolds

“…The study of the rationality of higher dimensional Fano manifolds is a very active area of research. Many new and interesting contributions and conjectures appeared in the last decades, mostly concerning the irrationality of very general Fano hypersurfaces (see for example the recent survey [Kol19] and the references therein). Deep recent contributions in [KT19] imply that the locus of geometrically rational fibers in a smooth family of projective manifolds is closed under specialisation, improving substantially our understanding of the loci of rational objects in the corresponding moduli spaces, see [HPT18] for very significative examples in dimension four.…”

Lines, conics, and all that

“…of a smooth Fano threefold X is a principally polarized abelian variety. Using a fine structure of the intermediate Jacobian one can detect the non-rationality of X; see, e.g., [33,106,271,374,457,473], cf. also [10,83,372,398].…”

About the cover: Quaste

“…Here, X, Y, Z can be algebraic or analytic varieties or germs thereof, and isomorphic may mean biregular, birational, biholomorphic, or just formal isomorphism. The answer depends very much on the context, with scattered results in the biregular case (small dimensions, smooth Z [1], [2]), counterexamples and various subtle theorems in the birational setting [3], [4], [5], [6], and rather complete positive results for local analytic varieties [7].…”