Measures of irrationality for hypersurfaces of large degree

Abstract

We study various measures of irrationality for hypersurfaces of large degree in projective space and other varieties. These include the least degree of a rational covering of projective space, and the minimal gonality of a covering family of curves. The theme is that positivity properties of canonical bundles lead to lower bounds on these invariants. In particular, we prove that if $X\subset \mathbb P^{n+1}$ is a very general smooth hypersurface of dimension $n$ and degree $d\ge 2n+1$, then any dominant rational mapping $f\colon X\dashrightarrow \mathbb P^n$ must have degree at least $d-1$. We also propose a number of open problems, and we show how our methods lead to simple new proofs of results of Ran and Beheshti–Eisenbud concerning varieties of multi-secant lines

The Konno invariant of some algebraic varieties

“…Recalling again that m i ≥ m 2 i , the required inequality follows. 1 In our setting, the vanishing in question is very elementary. In fact, the question being local, one can replace S by an affine neighborhood of x, so that O S ′ (−A) is globally generated.…”

On irrationality of hypersurfaces in $\mathbf {P}^{n+1}$

“…Now recall that we assume s ≥ 1. Then it follows from the computations of Ein and Voisin [4,Proposition 3.8] that e ≥ con.gon(S) ≥ d − 2n + s.…”

“…Therefore irr(X) = 1 if and only if X is rational. It was established in [3], [4] that if X ⊂ P n+1 is a very general smooth hypersurface of dimension n and degree d ≥ 2n + 1, then irr(X) = d − 1.…”

Degree of irrationality of very general abelian surfaces

“…The history behind the development of these ideas is described in [4]. The results of [2], [3], [4] depend on the positivity of the canonical bundles of the varieties in question, so it is interesting to consider what happens in the K X -trivial case. Our purpose here is to prove the somewhat surprising fact that the degree of irrationality of a very general polarized abelian surface is uniformly bounded above, independently of the degree of the polarization.…”