Abstract
AbstractLet F be a non-singular homogeneous polynomial of degree d in n variables. We give an asymptotic formula of the pairs of integer points $$(\mathbf {x}, \mathbf {y})$$
(
x
,
y
)
with $$|\mathbf {x}| \leqslant X$$
|
x
|
⩽
X
and $$|\mathbf {y}| \leqslant Y$$
|
y
|
⩽
Y
which generate a line lying in the hypersurface defined by F, provided that $$n > 2^{d-1}d^4(d+1)(d+2)$$
n
>
2
d
-
1
d
4
(
d
+
1
)
(
d
+
2
)
. In particular, by restricting to Zariski-open subsets we are able to avoid imposing any conditions on the relative sizes of X and Y.
Publisher
Springer Science and Business Media LLC
Reference25 articles.
1. Altman, A.B., Kleiman, S.L.: Foundations of the theory of Fano schemes. Compositio Math. 34(1), 3–47 (1977)
2. Batyrev, V.V., Tschinkel, Y.: Manin’s conjecture for toric varieties. J. Algebraic Geom. 7(1), 15–53 (1998)
3. Beheshti, R.: Linear subvarieties of hypersurfaces. Int. Math. Res. Not. 49, 3055–3063 (2005)
4. Beheshti, R.: Lines on projective hypersurfaces. J. Reine Angew. Math. 592, 1–21 (2006)
5. Beheshti, R.: Hypersurfaces with too many rational curves. Math. Ann. 360(3–4), 753–768 (2014)
Cited by
1 articles.
订阅此论文施引文献
订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献