论文标题
霍奇基因座的小型编码成分，包含fermat品种
Small codimension components of the Hodge locus containing the Fermat variety
论文作者
论文摘要
我们表征了光滑度$ d $ d $ hypersurfaces $ \ mathbb {p}^{n+1} $偶数$ n $的hodge基因座的最小的编纂组件。它们对应于包含尺寸的线性代数周期$ \ frac {n} {2} $的Hypersurfaces的轨迹。此外，我们证明，在与通过Fermat的非线性周期相关的所有本地Hodge基因座中，与$（1,1，\ ldots，1,2）的完整相交周期相关的循环相关联，这是其Zariski狭窄空间的最小可能的编成。这回答了Movasati的猜想，并概括了Voisin的结果，即Noether-Lefschetz locus的组成之间的第一个差距到任意维度，前提是它们包含Fermat品种。
We characterize the smallest codimension components of the Hodge locus of smooth degree $d$ hypersurfaces of the projective space $\mathbb{P}^{n+1}$ of even dimension $n$, passing through the Fermat variety (with $d\neq 3,4,6$). They correspond to the locus of hypersurfaces containing a linear algebraic cycle of dimension $\frac{n}{2}$. Furthermore, we prove that among all the local Hodge loci associated to a non-linear cycle passing through Fermat, the ones associated to a complete intersection cycle of type $(1,1,\ldots,1,2)$ attain the minimal possible codimension of their Zariski tangent spaces. This answers a conjecture of Movasati, and generalizes a result of Voisin about the first gap between the codimension of the components of the Noether-Lefschetz locus to arbitrary dimension, provided that they contain the Fermat variety.
