Algebraic hyperbolicity of very general surfaces

Coskun, Izzet; Riedl, Eric

doi:10.1007/s11856-022-2379-2

Isr. J. Math.

2022

DOI: 10.1007/s11856-022-2379-2

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Algebraic hyperbolicity of very general surfaces

Izzet Coskun

Eric Riedl

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“…Let 𝑍 𝑖 denote the closure of the locus in 𝑋 swept out by lines meeting 𝑋 in at most 𝑖 points. By results of Ein [12,13], Pacienza [18], Voisin [25,26] and the authors [9], a very general hypersurface of degree 𝑑 ⩾ 2𝑛 − 2 + max(0, 4 − 𝑛) is algebraically hyperbolic. When 𝑑 ⩽ 2𝑛 − 3, every hypersurface of degree 𝑑 contains lines, so 𝑋 cannot be algebraically hyperbolic.…”

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confidence: 96%

Clustered families and applications to Lang‐type conjectures

Coskun

Riedl

2022

Proceedings of London Math Soc

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We introduce and classify 1‐clustered families of linear spaces in the Grassmannian double-struckGfalse(k−1,nfalse)$\mathbb {G}(k-1,n)$ and give applications to Lang‐type conjectures. Let X⊂Pn$X \subset \mathbb {P}^n$ be a very general hypersurface of degree d$d$. Let ZL$Z_L$ be the locus of points contained in a line of X$X$. Let Z2$Z_2$ be the closure of the locus of points on X$X$ that are swept out by lines that meet X$X$ in at most 2 points. We prove that: if d⩾3n+22$d \geqslant \frac{3n+2}{2}$, then X$X$ is algebraically hyperbolic modulo ZL;$Z_L\hbox{\it ;}$ if d⩾3n2$d \geqslant \frac{3n}{2}$, X$X$ contains lines but no other rational curves; if d⩾3n+32$d \geqslant \frac{3n+3}{2}$, then the only points on X$X$ that are rationally Chow zero equivalent to points other than themselves are contained in Z2$Z_2$; if d⩾3n+22$d \geqslant \frac{3n+2}{2}$ and a relative Green–Griffiths–Lang Conjecture holds, then the exceptional locus for X$X$ is contained in Z2$Z_2$. These sharpen prior results of Ein, Voisin, Pacienza, Clemens and Ran.

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confidence: 96%

Clustered families and applications to Lang‐type conjectures

Coskun

Riedl

2022

Proceedings of London Math Soc

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Algebraic hyperbolicity of very general surfaces

Cited by 1 publication

References 19 publications

Clustered families and applications to Lang‐type conjectures

Clustered families and applications to Lang‐type conjectures

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