加权射影三维空间中超曲面的对称微分
Symmetric differentials on weighted hypersurfaces
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中文总结 AI 辅助
本文研究加权射影三维空间中超曲面的对称微分，证明光滑及拟光滑超曲面的消失性质，并构造具有大余切丛的奇异例子，结合同时消解得到对称多重亏格从零跳跃至三次增长的形变等价曲面。
中文摘要 AI 辅助
我们研究了加权射影三维空间中超曲面上的对称微分。对于加权次数 \\(d>\sum a_i\\)，我们证明了光滑超曲面的扭曲消失性质以及良构拟光滑超曲面相应的自反消失性质。在相反方向上，利用Asega--De Oliveira--Weiss的商奇点判据以及加权Kummer和Segre构造，我们构造了奇异加权超曲面，其极小消解具有大余切丛。这些例子包括对每个 \\(r\ge2\\) 在 \\(\PP(1,1,1,r)\\) 中的例子。将消失性和大性结果与同时消解相结合，得到了形变等价的光滑射影曲面，其对称多重亏格从每个正阶的零跳跃到三次渐近增长。
英文摘要
We study symmetric differentials on weighted hypersurfaces. Let $k$ be an algebraically closed field and let $$ X_d\subset \mathbb{P}_k(a_0,\ldots,a_N),\qquad N\ge3, $$ be a weighted hypersurface, with $a_{\min}:=\min_i a_i$. In characteristic zero we prove twisted vanishing in the range $t<m a_{\min}$ for quasi-smooth hypersurfaces of degree $d\ge2a_{\min}$, with the corresponding reflexive statement on the coarse space. We also prove a tame positive-characteristic analogue; in particular, reflexive symmetric differentials of every positive order vanish. In the second part we specialize to $k=\mathbb{C}$ and to surfaces in weighted projective three space. Using the quotient-singularity criterion of Asega--De Oliveira--Weiss together with weighted Kummer and Segre constructions, we produce singular weighted hypersurfaces whose minimal resolutions have big cotangent bundle. Combining the vanishing and bigness results with simultaneous resolution yields deformation-equivalent smooth projective surfaces for which the symmetric plurigenera jump from zero in every positive order to cubic asymptotic growth.
发表机构
- Shanghai Normal University（上海师范大学）
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