Eigenschemes of Ternary Tensors

Equations of tensor eigenschemes

“…We are now in position to characterize determinantal generators of symmetric tensors. The next result proves [3,Conjecture 3.13].…”

“…There are several challenging questions about eigenschemes, their dimension, their degree, their configurations and their defining equations. In this paper we extend some of the results obtained in [3] in the case n = 2 to arbitrary dimension. For the convenience of the reader, we summarize here our main contributions:…”

“…Remark 2.7. Notice that the defining ideal of an eigenscheme of a tensor, even of a symmetric tensor, may fail to be saturated, as shown in [3,Remark 3.2]. In this case the eigenscheme has dimension at least 1.…”

“…Namely, we would like a similar characterization for determinantal equations of symmetric tensors. In particular, we will prove [3,Conjecture 3.13]. Our argument relies on the exactness of the affine de Rham complex, which we briefly recall.…”

“…• In Theorem 2.6 we explicitly compute the minimal graded free resolution of the defining ideal of a general eigenscheme. • In Theorem 3.1 and Theorem 3.4 we characterize determinantal equations of eigenschemes of tensors and symmetric tensors, thereby giving a positive solution to [3,Conjecture 3.13].…”