Let $X$ be a K3 surface, and $H$ its primitive polarization of the degree$H^2=8$. The moduli space of sheaves over $X$ with the isotropic Mukai vector$(2,H,2)$ is again a K3 surface, $Y$. In math.AG/0206158 we gave necessary andsufficient conditions in terms of Picard lattice of $X$ when $Y$ is isomorphicto $X$. The proof of sufficient condition in math.AG/0206158, when $Y$ isisomorphic to $X$, used Global Torelli Theorem for K3 surfaces, and it was noteffective. Here we give an effective variant of these results: its sufficient part givesan explicit isomorphism between $Y$ and $X$. We hope that our similar results in math.AG/0304415, math.AG/0307355,math.AG/0309348 for arbitrary primitive isotropic Mukai vector on a K3 surfacealso can be made effective.
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机译:假设$ X $是K3曲面,而$ H $是其原始极化度$ H ^ 2 = 8 $。带各向同性Mukai向量$(2,H,2)$的滑轮X的模量空间又是K3曲面,即Y。在math.AG/0206158中,当$ Y $同构为$ X $时,我们根据$ X $的皮卡德晶格给出了必要和充分的条件。当$ Y $同构为$ X $时,math.AG / 0206158中充分条件的证明对K3曲面使用了全局Torelli定理,但无效。在这里,我们给出了这些结果的有效变体:其足够的部分给出了$ Y $和$ X $之间的显式同构。我们希望,对于在K3曲面上的任意原始各向同性Mukai矢量,在math.AG/0304415、math.AG/0307355、math.AG/0309348中的类似结果也可以有效。
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