## Geometric height inequalities (1996)

Venue: | Math. Res. Lett |

Citations: | 11 - 2 self |

### BibTeX

@ARTICLE{Liu96geometricheight,

author = {Kefeng Liu},

title = {Geometric height inequalities},

journal = {Math. Res. Lett},

year = {1996},

volume = {3},

pages = {693--702}

}

### OpenURL

### Abstract

0. The Results. Let f: X → B be a fibration of a compact smooth algebraic surface over a compact Riemann surface B, denote by g ≥ 2 the genus of a generic fiber of f and by q the genus of B. Let s be the number of singular fibers of f and ωX/B be the relative dualizing sheaf. Let C1, · · · , Cn be n mutually disjoint sections of f, and denote by D the divisor �n j=1 Cj. Then the main result we are going to prove in this note is the following Theorem 0.1 Theorem 0.1. If f is not isotrivial and semistable, then (ωX/B + D) 2 < (2g − 2 + n)(2q − 2 + s). In fact our proof shows that, for any n, (ωX/B + D) 2 = (2g − 2 + n)(2q − 2 + s). if and only if f is isotrivial. We can derive several corollaries from this theorem. To state the results, we first introduce some notations. Let k be the function field of B and ¯ k be its algebraic closure. For an algebraic point P ∈ X ( ¯ k), we let CP be the corresponding horizontal curve on X. Let hK(P) = ωX/B · CP [k(P) : k] , d(P) = 2g ( ˜ CP) − 2 [k(P) : k] be respectively the geometric height and the geometric logarithmic discriminant of P. Here ˜ CP is the normalization of CP and [k(P) : k] = F ·CP, where 1 F is a generic fiber of f, is the degree of P. Let bP be the number of ramification points on ˜ CP of the induced map r: ˜ CP → B. Write dP = [k(P) : k]. Then we have Theorem 0.2. If f is semistable and not isotrivial, then

### Citations

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Diophantine approximations and value distribution $the\mathrm{o}\mathrm{r}.)^{\gamma
- Vojta
- 1987
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Citation Context ...dification of our method can be used to prove the (1+ε) conjecture. For the long history of height inequalities and the importance of the Vojta conjecture, we refer the reader to [La], Chapter VI, or =-=[V2]-=-. Especially such inequalities immediately imply the Mordell conjecture for functional field. When n = 0, Theorem 0.1 has the following straightforward consequence [B], [Ta1], 2 dPsCorollary 0.5. (Bea... |

19 |
P.G.: A local index theorem for families of ∂-operators on punctured Riemann surfaces and a new Kähler metric on their moduli spaces
- Takhtajan, Zograf
- 1991
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Citation Context ...z-Yau Lemma. We will use the conventions in [W], [W1] for the geometry of moduli space of semistable curves. Let Mg,n be the moduli space of Riemann surfaces of genus g with n punctures. As in [W] or =-=[TZ]-=-, we can consider Mg,n as the moduli space of complex structures on a fixed n-punctured Riemann surface R. Let π : Tg,n → Mg,n be the universal curve and ¯π : ¯ Tg,n → ¯ Mg,n their Deligne-Mumford com... |

15 | The minimal number of singular fibers of a semistable curve over P 1 - Tan - 1995 |

11 |
Number Theory III
- Lang
- 1991
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Citation Context ...ity. I hope that a modification of our method can be used to prove the (1+ε) conjecture. For the long history of height inequalities and the importance of the Vojta conjecture, we refer the reader to =-=[La]-=-, Chapter VI, or [V2]. Especially such inequalities immediately imply the Mordell conjecture for functional field. When n = 0, Theorem 0.1 has the following straightforward consequence [B], [Ta1], 2 d... |

11 | On algebraic points on curves - Vojta - 1991 |

6 |
On the invariants of base changes of pencils of curves
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Citation Context ...er to [La], Chapter VI, or [V2]. Especially such inequalities immediately imply the Mordell conjecture for functional field. When n = 0, Theorem 0.1 has the following straightforward consequence [B], =-=[Ta1]-=-, 2 dPsCorollary 0.5. (Beauville Conjecture) If B = CP 1 and f is semistable and not isotrivial, then s ≥ 5. Recall that f is called a Kodaira fibration, if f is everywhere of maximal rank but not a c... |

5 | Height inequality of algebraic points on curves over functional fields
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- 1994
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Citation Context ...ctured that the above inequality holds with (2 + ε) replaced by (1 + ε). Take n = 1 in Theorem 0.1, we get the following geometric height inequality, a weaker version of which was fisrt proved by Tan =-=[Ta]-=-. For simplicity we still assume f is semistable, the general case follows from the semistable reduction trick as used in [Ta]. Corollary 0.4. Assume f is not isotrivial, then hK(P ) ≤ (2g − 1)(d(P ) ... |

4 |
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- Vojta
- 1988
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Citation Context ... f is semistable and not isotrivial, then 2g − 2 hK(P ) < (1 + )(d(P ) + bP + s) − ω2 X/B . dP This theorem gives us a corollary about the geometric height inequality which is originally due to Vojta =-=[V]-=-. Corollary 0.3. Given any ε > 0, there exists a constant Oε(1) depending on ε, s, g and q, such that dP hK(P ) ≤ (2 + ε)d(P ) + Oε(1). Vojta conjectured that the above inequality holds with (2 + ε) r... |

1 |
Le nombre minimum de fibres singullieres D’une courbe stable sur
- Beauville
- 1981
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Citation Context ... reader to [La], Chapter VI, or [V2]. Especially such inequalities immediately imply the Mordell conjecture for functional field. When n = 0, Theorem 0.1 has the following straightforward consequence =-=[B]-=-, [Ta1], 2 dPsCorollary 0.5. (Beauville Conjecture) If B = CP 1 and f is semistable and not isotrivial, then s ≥ 5. Recall that f is called a Kodaira fibration, if f is everywhere of maximal rank but ... |

1 |
Harmonic maps and the geometry of Teichmuller spaces
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- 1991
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Citation Context ...proved. His method is algebro-geometric and is completely different from that of [Liu]. The key technique in [Liu] is the Schwarz-Yau lemma [Y] and the curvature computations of Wolpert and Jost [W], =-=[J]-=- for the WeilPeterson metric on the moduli spaces of curves. Theorem 0.2 follows from Theorem 0.1 and the stablization theorem in [Kn]. This work is partially supported by an NSF grant. I would like t... |

1 |
The projectivity of the moduli space of stable curves
- Knutson
- 1983
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Citation Context ... lemma [Y] and the curvature computations of Wolpert and Jost [W], [J] for the WeilPeterson metric on the moduli spaces of curves. Theorem 0.2 follows from Theorem 0.1 and the stablization theorem in =-=[Kn]-=-. This work is partially supported by an NSF grant. I would like to thank S. Lu, S. Tan, I. Tsai and S. T Yau for helpful discussions. 1. Moduli Spaces and the Schwarz-Yau Lemma. We will use the conve... |

1 |
Remarks on the geometry of moduli spaces, Harvard preprint
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- 1991
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Citation Context ... particularly implies that a Kodaira surface can not be uniformized by a ball. I was told by Tan and Tsai that this has been unknown for a long time. The method to prove Theorem 0.1 was first used in =-=[Liu]-=- to prove the case of n = 0. In [Ta1], Corollary 0.5 was proved by establishing a weaker version of the n = 0 case of Theorem 0.1. Also in [Ta], a weaker version of Corollary 0.4 is proved. His method... |