Fano Schemes of Determinants and Permanents

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Gives a precise criterion for when the Fano scheme F_k(D_n) of k-planes in the determinant hypersurface is connected: components come from compression space components and connectivity reduces to which torus-fixed points lie in which compression spaces, proved by torus action and tangent-space/deformation calculations. Key exact facts: F1(D_n) has exactly n irreducible components each of dimension 2n^2 − n − 1; maximal linear spaces have dimension n(n−1). Partial analogues for the permanent show many more components (e.g. 21 vs 3) and yield a product-rank lower bound per_3 = 4.

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1. 0:00Introduction
2. 0:28Singular matrix spaces
3. 1:23Compression spaces
4. 3:54Complexity theory context
5. 5:48Fano schemes defined
6. 7:24Planes of 2x2 matrices
7. 9:30Geometry of Fano schemes
8. 12:44Higher dimensional spaces
9. 15:27Connectedness results
10. 19:37Proof sketch
11. 24:05Proving disconnectedness
12. 27:28Permanents vs determinants
13. 32:35Applications and open problems