In this very interesting paper the authors introduce new Grassmannian techniques that are applied towards Lang-type conjectures. Let us recall that if \(X\) is a variety of general type, then Lang conjectures that there exists a proper subvariety \(Z \subset X\) such that
1) the images of non-constant maps from rational curves and abelian varieties into \(X\) are contained in \(Z\);
2) the images of non-constant entire curves are contained in \(Z\);
3) the complement of \(Z\) is Kobayashi hyperbolic.
Furthermore, Lang predicts that these geometric conditions control the arithmetic of the variety \(X\) and he conjectures that if \(X\) is defined over a number field \(K\) and \(L\) is an algebraic extension of \(K\), then \(X \setminus Z(L)\) is finite.
Recall that a projective variety \(Y\) is algebraically hyperbolic if there exists \(\varepsilon >0\) such that any reduced and connected curve \(C \subset Y\) of geometric genus \(g(C)\) satisfies \[ 2g(C) - 2 \geq \varepsilon \cdot\mathrm{deg}(C).\tag{\(\star\)} \] Moreover, we say that \(Y\) is algebraically hyperbolic modulo \(Z\) if the inequality \((\star)\) holds for any curve \(C\) not contained in \(Z\).
The first result of the paper is devoted to the algebraic hyperbolicity for very general hypersurfaces \(X \subset \mathbb{P}^{n}\) of degree \(d\) with \(n\geq 3\). Let us denote by \(Z_{L}\) the locus of points contained in a line of \(X\) and we denote by \(Z_{i}\) the closure of the locus in \(X\) swept out by lines meeting \(X\) in at most \(i\) points.
Theorem A. If \(d \geq \frac{3n+2}{2}\), then any curve not lying in \(Z_{L}\) satisfies \(2g(C)-2 \geq\mathrm{deg}(C)\), where \(g(C)\) is the geometric genus of \(C\). In particular, \(X\) is algebraically hyperbolic modulo \(Z_{L}\).
The next result is devoted to rationally Chow-0 equivalent points (so points on \(X\) that lie on rational curves).
Theorem B. Let \(X\) be a very general hypersurface in \(\mathbb{P}^{n}\) of degree \(d\).
1) Let \(k\) be a positive integer. If \(d \geq \frac{3n+1-k}{2}\), then the only points of \(X\) rationally equivalent to a \(k\)-dimensional family of points other than themselves are this that lie in \(Z_{1}\).
2) If \(d \geq \frac{3n}{2}\), then \(X\) contains lines but no other rational curves.
3) If \(d \geq \frac{3n+3}{2}\), then any point on \(X\) rationally equivalent to another point of \(X\) lies in \(Z_{2}\).
The final application is devoted to the exceptional set in the Green-Griffiths-Lang Conjecture. This conjecture predicts that if a variety \(Y\) is of general type, then the images of all non-constant entire curves are contained in a proper algebraic subvariety.
Theorem C. If \(d \geq \frac{3n+2}{2}\) and a relative version of the Green-Griffiths-Lang Conjecture holds, then the exceptional locus for \(X\) is contained in \(Z_{2}\).
The main techniques developed by the authors, in order to show the above results, are devoted to the so-called \(\ell\)-clustered families of subspaces in the Grassmannians. Let \(B \subset \mathbb{G}(k-1,n)\) denote an irreducible family of \((k-1)\)-dimensional projective linear spaces in \(\mathbb{P}^{n}\). Assume that the codimension of \(B\) is \(\varepsilon >0\). Let \(C \subset \mathbb{G}(k,n)\) denote the family of \(k\)-dimensional projective linear spaces consisting of those linear spaces that contain a member of \(B\). One calls \(C\) the containing family of \(B\). The codimension of \(C\) in \(\mathbb{G}(k,n)\) is at most \(\varepsilon -1\). The family \(B\) is \(\ell\)-clustered if the codimension of \(C\) in \(\mathbb{G}(k,n)\) is \(\varepsilon - \ell\). The main contribution revolving around clustered families is devoted to the classification of \(1\)-clustered families in \(\mathbb{G}(k-1,n)\).