论文标题
阳性特征的相对规范除数和不释放性的伪效应
Pseudo-effectivity of the relative canonical divisor and uniruledness in positive characteristic
论文作者
论文摘要
我们表明，如果$ f \ colon x \ to t $是在代数封闭的特征性特征性$ p> 0 $的光滑投射品种之间的旋转形态，则具有几何积分且非释放的通用纤维，然后$ k_ {x/t} $是Pseudo-extective-pseudo-extective。 证明是基于覆盖有理曲线的$ x $的，这在基本和通用纤维都不含量时就会产生矛盾。但是，我们仅假定通用纤维不是未释放的。因此，最难证明的部分是表明，基地的平滑不平稳的封面我们显示以下内容：如果$ t $是$ k $的平稳射击品种，而$ \ nmatcal {a} $是足够的线条套件，则是$ p \ nmid d $的循环封面，然后由$ p \ nmid d $ wefter | wefter p \ y \ y \ y \ y \ y \ y \ y \ y \ y \ y \ y \ y \ y \ y \ y \ y \ y \ y \ y \ y \ y \ y |不是不释放的。为此，我们显示了以下共同体学条件，这可能具有独立的兴趣：每当$ h^n（t，\ mathcal {o} _t）的半稳定部分的尺寸大于$ h^{n-1-1}（n n-1-1}（n n-1-1}（t，t，t，t，t，t，t，t，t，t，t，t，t，t，t，t，t，t，t，n $）$ h^n（t，\ natercal {o} _t）$都大于$ h^n（t，\ natercal {o} _t）$时，平稳的投射变化$ t $ t $ t $。 此外，我们还显示上述所有语句的单数版本。
We show that if $f\colon X \to T$ is a surjective morphism between smooth projective varieties over an algebraically closed field $k$ of characteristic $p>0$ with geometrically integral and non-uniruled generic fiber, then $K_{X/T}$ is pseudo-effective. The proof is based on covering $X$ with rational curves, which gives a contradiction as soon as both the base and the generic fiber are not uniruled. However, we assume only that the generic fiber is not uniruled. Hence, the hardest part of the proof is to show that there is a finite smooth non-uniruled cover of the base for which we show the following: If $T$ is a smooth projective variety over $k$ and $\mathcal{A}$ is an ample enough line bundle, then a cyclic cover of degree $p \nmid d$ given by a general element of $\left|\mathcal{A}^d\right|$ is not uniruled. For this we show the following cohomological uniruledness condition, which might be of independent interest: A smooth projective variety $T$ of dimenion $n$ is not uniruled whenever the dimension of the semi-stable part of $H^n(T, \mathcal{O}_T)$ is greater than that of $H^{n-1}(T, \mathcal{O}_T)$. Additionally, we also show singular versions of all the above statements.
