论文标题
投影束和弱Zariski分解的伪有效锥
Pseudo-effective cones of projective bundles and weak Zariski decomposition
论文作者
论文摘要
在本文中，我们考虑了光滑的复杂的投影型$ x $上的投影捆绑包$ \ mathbb {p} _x（e）$，其中$ e $是$ x $的半套件，$ c_2（end（e））= 0 $。我们提供了必要和足够的条件以获得平等$ nef^1 \ bigl（\ MathBb {p} _x（e）\ bigr）= \ edimline {eff}^1 \ bigl（\ mathbb {p} _x（e） $ \ mathbb {p} _x（e）$。为了应用结果，我们显示了一些特殊品种的投影捆绑包的Nef和伪芬锥的平等。特别是，我们表明Zariski分解薄弱是在这些投射束上的。我们还表明，在平滑的复杂的PICARD数字1 is $ k $ - homosiene的$ k $ homosous，即$ k $ \ operline {eff}^k \ bigl（big bigl（big big big big big big big）上， nef^k \ bigl（\ Mathbb {p} _ {x}（e）\ bigr）$ for $ 1 \ leq k <r $。最后，我们表明，对于光滑的投影曲线$ c $，纤维产品$ \ mathbb {p} _c（e）\ times_c \ times_c \ mathbb {p}（e'）$。
In this article, we consider the projective bundle $\mathbb{P}_X(E)$ over a smooth complex projective variety $X$, where $E$ is a semistable bundle on $X$ with $c_2(End(E)) =0$. We give a necessary and sufficient condition to get the equality $ Nef^1\bigl(\mathbb{P}_X(E)\bigr) = \overline{Eff}^1\bigl(\mathbb{P}_X(E)\bigr)$ of nef cone and pseudoeffective cone of divisors in $\mathbb{P}_X(E)$. As an application of our result, we show the equality of nef and pseudoeffective cones of divisors of projective bundles over some special varieties. In particular, we show that weak Zariski decomposition exists on these projective bundles. We also show that a semistable bundle $E$ of rank $r \geq 2$ with $c_2\bigl(End(E)\bigr) = 0$ on a smooth complex projective variety of Picard number 1 is $k$-homogeneous i.e. $\overline{Eff}^k\bigl(\mathbb{P}_X(E)\bigr) = Nef^k\bigl(\mathbb{P}_{X}(E)\bigr)$ for all $1 \leq k < r$. Finally, we show that weak Zariski decomposition exists for a fibre product $\mathbb{P}_C(E)\times_C\mathbb{P}(E')$ over a smooth projective curve $C$.
