论文标题
ODECO张量的二次方程式
On the quadratic equations for odeco tensors
论文作者
论文摘要
Elina Robeva发现了正交分解（“ ODECO”）张量满足的二次方程。然后，Boralevi-Draisma-Horobeţ-Robeva随后证明，这些方程式是ODECO张量的特征。这就提出了一个问题在多大程度上还表征了ODECO张量的Zariski闭合量，而不是复数。在当前的论文中，我们将自己限制在对称订单第三订单的张量，即格式$ n \ times n \ times n $。通过为Robeva的猜想提供明确的反例，我们表明，对于$ n \ geq 12 $，这些方程式不够。此外，在开放子集中，张量的切片的线性跨度包含一个可逆矩阵，我们表明Robeva的方程式削减了dimension $ n \ leq 13 $的odeco张量的极限，而不是$ n \ geq 14 $。为此，我们表明Robeva的方程式基本上捕获了$ n $点的希尔伯特计划中的Gorenstein Locus，并且我们使用Casnati-Jelisiejew-Notari的作品来降低此基因座的（IR）降低。
Elina Robeva discovered quadratic equations satisfied by orthogonally decomposable ("odeco") tensors. Boralevi-Draisma-Horobeţ-Robeva then proved that, over the real numbers, these equations characterise odeco tensors. This raises the question to what extent they also characterise the Zariski-closure of the set of odeco tensors over the complex numbers. In the current paper we restrict ourselves to symmetric tensors of order three, i.e., of format $n \times n \times n$. By providing an explicit counterexample to one of Robeva's conjectures, we show that for $n \geq 12$, these equations do not suffice. Furthermore, in the open subset where the linear span of the slices of the tensor contains an invertible matrix, we show that Robeva's equations cut out the limits of odeco tensors for dimension $n \leq 13$, and not for $n \geq 14$ on. To this end, we show that Robeva's equations essentially capture the Gorenstein locus in the Hilbert scheme of $n$ points and we use work by Casnati-Jelisiejew-Notari on the (ir)reducibility of this locus.
