Secant varieties of flag varieties via Schur apolarity

@misc{bernardi2026secant,
  title = {{Secant varieties of flag varieties via Schur apolarity}},
  author = {Alessandra Bernardi and Stefano Canino and Vincenzo Antonio Isoldi},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.09301},
  eprint = {2609.09301},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.09301},
}

TY  - UNPB
TI  - Secant varieties of flag varieties via Schur apolarity
AU  - Alessandra Bernardi
AU  - Stefano Canino
AU  - Vincenzo Antonio Isoldi
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.09301
UR  - https://arxiv.org/abs/2609.09301
AB  - We develop a general first-order theory of Schur apolarity for the study of secant varieties of flag varieties in arbitrary homogeneous embeddings. Extending the classical apolarity--fat-point correspondence for Veronese varieties, we show that in the Schur setting the algebraic square of the apolar ideal need not coincide with the geometric double-point conditions. We introduce a geometric Schur square whose relevant component is the conormal space, yielding a Schur Dual Terracini Lemma. Our construction recovers classical apolarity in the symmetric case. A slot-by-slot Consistency Theorem realizes these intrinsic conditions as multigraded double points. As an application, we determine the dimensions of all secant varieties of Fl(1,2;V_n) embedded by 𝒪(1,1): the only defective cases are σ_2(Fl(1,2;V_3)) and σ_3(Fl(1,2;V_4)), both of defect one.
ER  -