Related papers: Restrictions on rational surfaces lying in very ge…

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…

We show that complex Fano hypersurfaces can have arbitrarily large degrees of irrationality. More precisely, if we fix a Fano index e, then the degree of irrationality of a very general complex Fano hypersurface of index e and dimension n…

A very general hypersurface of dimension $n$ and degree $d$ in complex projective space is rational if $d \leq 2$, but is expected to be irrational for all $n, d \geq 3$. Hypersurfaces in weighted projective space with degree small relative…

This paper treats the dominant rational maps from the product of two very general curves to nonsingular projective surfaces. Combining the result by Bastianelli and Pirola, we prove that the product of two very general curves of genus…

We prove that a very general complex hypersurface of degree $n+1$ in $\mathbb{P}^{n+1}$ containing an $r$-plane with multiplicity $m$ is not stably rational for $n \ge 3$, $m, r > 0$ and $n \ge m+r$. We also investigate failure of stable…

We give the first evidence for a conjecture that a general, index-one, Fano hypersurface is not unirational: (i) a general point of the hypersurface is contained in no rational surface ruled, roughly, by low-degree rational curves, and (ii)…

We study rational curves on general Fano hypersurfaces in projective space, mostly by degenerating the hypersurface along with its ambient projective space to reducible varieties. We prove results on existence of low-degree rational curves…

We study the dominant rational maps from a general surface in P^{3} to surfaces of general type. We prove restrictions on the target surfaces, and special properties of the rational maps. We show that for a small degree the general surface…

Koll\'ar proved that a very general $n$-dimensional complex hypersurface of degree at least $3\lceil (n+3)/4\rceil$ is not birational to a fibration in rational curves. This is most interesting when the hypersurface is Fano, in which case…

We show that the degrees of rational endomorphisms of very general complex Fano and Calabi-Yau hypersurfaces satisfy certain congruence conditions by specializing to characteristic p. As a corollary we show that very general n-dimensional…

We prove that rationally essential manifolds with suitably large fundamental groups do not admit any maps of non-zero degree from products of closed manifolds of positive dimension. Particular examples include all manifolds of non-positive…

We prove that a very general projective K3 surface does not admit a dominant self rational map of degree at least two.

Let $X$ be a projective variety and let $C$ be a rational normal curve on $X$. We compute the normal bundle of $C$ in a general complete intersection of hypersurfaces of sufficiently large degree in $X$. As a result, we establish the…

In this paper we investigate the existence of generically finite dominant rational maps from products of curves to surfaces of general type. We prove that the product CxD of two distinct very general curves of genus g>6 and g'>1 does not…

In this paper we study birational immersions from a very general smooth plane curve to a non-rational surface with $p_g=q=0$ to treat dominant rational maps from a very general surface $X$ of degree$\geq 5$ in ${\mathbb P}^3$ to smooth…

In this paper we try to further explore the linear model of the moduli of rational maps. Our attempt yields following results. Let $X\subset \mathbf P^n$ be a generic hypersurface of degree $h$. Let $R_d(X, h)$ denote the open set of the…

The families of smooth rational surfaces in $\PP^4$ have been classified in degree $\le 10$. All known rational surfaces in $\PP^4$ can be represented as blow-ups of the plane $\PP^2$. The fine classification of these surfaces consists of…

The purpose of this note is to show that the minimal $e$ for which every smooth Fano hypersurface of dimension $n$ contains a free rational curve of degree at most $e$ cannot be bounded by a linear function in $n$ when the base field has…

Fix positive integers $n,r,d$. We show that if $n,r,d$ satisfy a suitable inequality, then any smooth hypersurface $X\subset \mathbb{P}^n$ defined over a finite field of characteristic $p$ sufficiently large contains a rational $r$-plane.…

We show that a wide class of hypersurfaces in all dimensions are not stably rational. Namely, for all d at least about 2n/3, a very general complex hypersurface of degree d in P^{n+1} is not stably rational. The statement generalizes…