多重Fujita逼近
Multigraded Fujita Approximation
摘要：
原始的Fujita逼近定理指出，射影簇$X$上的大因子$D$的体积总是可以任意地逼近于$X$的双有理修改上的充分因子的自交数.也可以用分次线性级数来表示它：令$W_{\bullet} = \{W_k \}$是与大除数$D$相关的完全分次线性级数：\[ W_k = H^0\big(X,\mathcal{O}_X(kD)\big). \]对于每个固定的正整数$p$，定义$W^{(p)}_{\bullet}$是由$W_p$生成的$W_{\bullet}$的分次线性子级数：\[ W^{(p)}_{m}={cases} 0, &\text{if $p\nmid m$;} \mathrm{Image} \big(S^k W_p \rightarrow W_{kp} \big), &\text{if $m=kp$.} {cases} \]然后$W^{(p)}_{\bullet}$的体积接近$W_{\bullet}$的体积为$p\to\infty$。我们将证明，在这个公式下，藤田逼近定理可以推广到多级线性级数的情况。
英文摘要：
The original Fujita approximation theorem states that the volume of a big divisor $D$ on a projective variety $X$ can always be approximated arbitrarily closely by the self-intersection number of an ample divisor on a birational modification of $X$. One can also formulate it in terms of graded linear series as follows: let $W_{\bullet} = \{W_k \}$ be the complete graded linear series associated to a big divisor $D$: \[ W_k = H^0\big(X,\mathcal{O}_X(kD)\big). \] For each fixed positive integer $p$, define $W^{(p)}_{\bullet}$ to be the graded linear subseries of $W_{\bullet}$ generated by $W_p$: \[ W^{(p)}_{m}={cases} 0, &\text{if $p\nmid m$;} \mathrm{Image} \big(S^k W_p \rightarrow W_{kp} \big), &\text{if $m=kp$.} {cases} \] Then the volume of $W^{(p)}_{\bullet}$ approaches the volume of $W_{\bullet}$ as $p\to\infty$. We will show that, under this formulation, the Fujita approximation theorem can be generalized to the case of multigraded linear series.
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