This table tries to give the state of the art of the ongoing project of understanding K-stability for all Fano 3-folds. Let me know if I have made a mistake, or am missing some recent progress!
Below the table we provide an overview of the open cases.
| ID | description | moduli | $\mathrm{Aut}^0$ | K‑stability | K‑polystability | K‑semistability | references | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1-1 | double cover of $\mathbb{P}^3$ with branch locus a divisor of degree 6
| 68 | $0$ | every member is K‑stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 1-2 |
|
| $0$ | every member is K‑stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 1-3 | complete intersection of quadric and cubic in $\mathbb{P}^5$ | 34 | $0$ | every member is K‑stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 1-4 | complete intersection of 3 quadrics in $\mathbb{P}^6$ | 27 | $0$ | every member is K‑stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 1-5 | Gushel–Mukai 3-fold
|
| $0$ | every member is K‑stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 1-6 | section of half-spinor embedding of a connected component of $\mathrm{OGr}_+(5,10)$ by codimension 7 subspace | 18 | $0$ | every member is K‑stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 1-7 | section of Plücker embedding of $\mathrm{Gr}(2,6)$ by codimension 5 subspace | 15 | $0$ | every member is K‑stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 1-8 | section of Plücker embedding of $\mathrm{SGr}(3,6)$ by codimension 3 subspace | 12 | $0$ | every member is K‑stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 1-9 | section of the adjoint $\mathrm{G}_2$-Grassmannian $\mathrm{G}_2\mathrm{Gr}(2,7)$ by codimension 2 subspace | 10 | $0$ | general member is K‑stable | general member is K‑polystable | general member is K‑semistable | |||||||||
| 1-10 | zero locus of $(\bigwedge^2\mathcal{U}^\vee)^{\oplus 3}$ on $\mathrm{Gr}(3,7)$ | 6 |
| general member is K‑stable but there exist members that are not | general member is K‑polystable but there exist members that are not | general member is K‑semistable | |||||||||
| 1-11 | double Veronese cone hypersurface of degree 6 in $\mathbb{P}(1,1,1,2,3)$ | 34 | $0$ | every member is K‑stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 1-12 | quartic double solid double cover of $\mathbb{P}^3$ with branch locus a smooth quartic surface
| 19 | $0$ | every member is K‑stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 1-13 | hypersurface of degree 3 in $\mathbb{P}^4$ | 10 | $0$ | every member is K‑stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 1-14 | complete intersection of 2 quadrics in $\mathbb{P}^5$ | 3 | $0$ | every member is K‑stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 1-15 | quintic del Pezzo threefold section of Plücker embedding of $\mathrm{Gr}(2,5)$ by codimension 3 subspace | 0 | $\mathrm{PGL}_2$ | none are K-stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 1-16 | hypersurface of degree 2 in $\mathbb{P}^4$ | 0 | $\mathrm{PSO}_5$ | none are K-stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 1-17 | projective space $\mathbb{P}^3$ | 0 | $\mathrm{PGL}_4$ | none are K-stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 2-1 | blowup of 1-11 in an elliptic curve which is the intersection of two divisors from half the anticanonical linear system | 36 | $0$ | every member is K‑stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 2-2 | double cover of 2-34 with branch locus a $(2,4)$-divisor | 33 | $0$ | every member is K‑stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 2-3 | blowup of 1-12 in an elliptic curve which is the intersection of two divisors from half the anticanonical linear system | 23 | $0$ | every member is K‑stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 2-4 | blowup of 1-17 in the intersection of two cubics
| 21 | $0$ | every member is K‑stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 2-5 | blowup of 1-13 in a plane cubic | 16 | $0$ | general member is K‑stable | general member is K‑polystable | general member is K‑semistable | |||||||||
| 2-6 | Verra 3-fold
|
| $0$ | every member is K‑stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 2-7 | blowup of 1-16 in the intersection of two divisors from $|\mathcal{O}_Q(2)|$ | 14 | $0$ | every member is K‑stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 2-8 |
| $0$ | every member is K‑stable | every member is K‑polystable | every member is K‑semistable | ||||||||||
| 2-9 | complete intersection of degree $(1,1)$ and $(2,1)$ in $\mathbb{P}^3\times\mathbb{P}^2$
| 13 | $0$ | general member is K‑stable | general member is K‑polystable | general member is K‑semistable | |||||||||
| 2-10 | blowup of 1-14 in an elliptic curve which is an intersection of 2 hyperplanes | 11 | $0$ | general member is K‑stable | general member is K‑polystable | general member is K‑semistable | |||||||||
| 2-11 | blowup of 1-13 in a line | 12 | $0$ | general member is K‑stable | general member is K‑polystable | general member is K‑semistable | |||||||||
| 2-12 | intersection of 3 $(1,1)$-divisors in $\mathbb{P}^3\times\mathbb{P}^3$
| 9 | $0$ | general member is K‑stable | general member is K‑polystable | general member is K‑semistable | |||||||||
| 2-13 | blowup of 1-16 in a curve of degree 6 and genus 2 | 8 | $0$ | general member is K‑stable | general member is K‑polystable | general member is K‑semistable | |||||||||
| 2-14 | blowup of 1-15 in an elliptic curve which is an intersection of 2 hyperplanes | 7 | $0$ | general member is K‑stable | general member is K‑polystable | general member is K‑semistable | |||||||||
| 2-15 |
| $0$ | every member is K‑stable | every member is K‑polystable | every member is K‑semistable | ||||||||||
| 2-16 | blowup of 1-14 in a conic | 7 | $0$ | general member is K‑stable | general member is K‑polystable | general member is K‑semistable | |||||||||
| 2-17 | blowup of 1-16 in an elliptic curve of degree 5 | 5 | $0$ | general member is K‑stable | general member is K‑polystable | general member is K‑semistable | |||||||||
| 2-18 | double cover of 2-34 with branch locus a divisor of degree $(2,2)$ | 6 | $0$ | every member is K‑stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 2-19 | blowup of 1-14 in a line | 5 | $0$ | general member is K‑stable | general member is K‑polystable | general member is K‑semistable | |||||||||
| 2-20 | blowup of 1-15 in a twisted cubic | 3 |
| general member is K‑stable but there exist members that are not | general member is K‑polystable but there exist members that are not | general member is K‑semistable | |||||||||
| 2-21 | blowup of 1-16 in a twisted quartic | 2 |
| general member is K‑stable but there exist members that are not | general member is K‑polystable but there exist members that are not | general member is K‑semistable | |||||||||
| 2-22 | blowup of 1-15 in a conic | 1 |
| general member is K‑stable but there exist members that are not | general member is K‑polystable but there exist members that are not | every member is K‑semistable | |||||||||
| 2-23 |
| $0$ | none are K-stable | none are K-polystable | none are K-semistable | ||||||||||
| 2-24 | divisor on $\mathbb{P}^2\times\mathbb{P}^2$ of bidegree $(1,2)$ | 1 |
| general member is K‑stable but there exist members that are not | general member is K‑polystable but there exist members that are not | every member is K‑semistable | |||||||||
| 2-25 | blowup of 1-17 in an elliptic curve which is an intersection of 2 quadrics
| 1 | $0$ | every member is K‑stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 2-26 | blowup of 1-15 in a line | 0 |
| none are K-stable | none are K-polystable | general member is K‑semistable but there exist members that are not | |||||||||
| 2-27 | blowup of 1-17 in a twisted cubic | 0 | $\mathrm{PGL}_2$ | none are K-stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 2-28 | blowup of 1-17 in a plane cubic | 1 | $\mathbb{G}_{\mathrm{a}}^3\rtimes\mathbb{G}_{\mathrm{m}}$ | none are K-stable | none are K-polystable | none are K-semistable | |||||||||
| 2-29 | blowup of 1-16 in a conic | 0 | $\mathbb{G}_{\mathrm{m}}\times\mathrm{PGL}_2$ | none are K-stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 2-30 | blowup of 1-17 in a conic | 0 | $\mathrm{PSO}_{5;1}$ | none are K-stable | none are K-polystable | none are K-semistable | |||||||||
| 2-31 | blowup of 1-16 in a line | 0 | $\mathrm{PSO}_{5;2}$ | none are K-stable | none are K-polystable | none are K-semistable | |||||||||
| 2-32 | divisor on $\mathbb{P}^2\times\mathbb{P}^2$ of bidegree $(1,1)$
| 0 | $\mathrm{PGL}_3$ | none are K-stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 2-33 | blowup of 1-17 in a line | 0 | $\mathrm{PGL}_{4;2}$ | none are K-stable | none are K-polystable | none are K-semistable | |||||||||
| 2-34 | $\mathbb{P}^1\times\mathbb{P}^2$ | 0 | $\mathrm{PGL}_2\times\mathrm{PGL}_3$ | none are K-stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 2-35 | $\mathrm{Bl}_p\mathbb{P}^3$
| 0 | $\mathrm{PGL}_{4;1}$ | none are K-stable | none are K-polystable | none are K-semistable | |||||||||
| 2-36 | $\mathbb{P}(\mathcal{O}_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}(2))$ | 0 | $\mathrm{Aut}(\mathbb{P}(1,1,1,2))$ | none are K-stable | none are K-polystable | none are K-semistable | |||||||||
| 3-1 | double cover of 3-27 with branch locus a divisor of degree $(2,2,2)$ | 17 | $0$ | every member is K‑stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 3-2 | divisor from $|\mathcal{L}^{\otimes 2}\otimes\mathcal{O}(2,3)|$ on the $\mathbb{P}^2$-bundle $\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(-1,-1)^{\oplus 2})$ over $\mathbb{P}^1\times\mathbb{P}^1$ such that $X\cap Y$ is irreducible, and $\mathcal{L}$ is the tautological bundle, and $Y\in|\mathcal{L}|$ | 11 | $0$ | general member is K‑stable | general member is K‑polystable | general member is K‑semistable | |||||||||
| 3-3 | divisor on $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^2$ of degree $(1,1,2)$ | 9 | $0$ | every member is K‑stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 3-4 | blowup of 2-18 in a smooth fiber of the composition of the projection to $\mathbb{P}^1\times\mathbb{P}^2$ with the projection to $\mathbb{P}^2$ of the double cover with the projection | 8 | $0$ | every member is K‑stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 3-5 | blowup of 2-34 in a curve $C$ of degree $(5,2)$ such that $C\hookrightarrow\mathbb{P}^1\times\mathbb{P}^2\to\mathbb{P}^2$ is an embedding | 5 |
| general member is K‑stable but there exist members that are not | general member is K‑polystable but there exist members that are not | general member is K‑semistable | |||||||||
| 3-6 | blowup of 1-17 in the disjoint union of a line and an elliptic curve of degree 4
| 5 | $0$ | general member is K‑stable | general member is K‑polystable | general member is K‑semistable | |||||||||
| 3-7 | blowup of 2-32 in an elliptic curve which is the intersection of two divisors from $|-\frac{1}{2}\mathrm{K}_W|$ | 4 | $0$ | general member is K‑stable | general member is K‑polystable | general member is K‑semistable | |||||||||
| 3-8 | divisor from the linear system $|(\alpha\circ\pi_1)^*(\mathcal{O}_{\mathbb{P}^2}(1)\otimes\pi_2^*(\mathcal{O}_{\mathbb{P}^2}(2))|$ where $\pi_i\colon\mathrm{Bl}_1\mathbb{P}^2\times\mathbb{P}^2\to\mathrm{Bl}_1\mathbb{P}^2,\mathbb{P}^2$ are the projections, and $\alpha\colon\mathrm{Bl}_1\mathbb{P}^2\to\mathbb{P}^2$ is the blowup | 3 |
| general member is K‑stable but there exist members that are not | general member is K‑polystable but there exist members that are not | general member is K‑semistable | |||||||||
| 3-9 | blowup of the cone over the Veronese of $\mathbb{P}^2$ in $\mathbb{P}^5$ with center the disjoint union of the vertex and a quartic curve on $\mathbb{P}^2$ | 6 | $\mathbb{G}_{\mathrm{m}}$ | none are K-stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 3-10 | blowup of 1-16 in the disjoint union of 2 conics
| 2 |
| general member is K‑stable but there exist members that are not | general member is K‑polystable but there exist members that are not | every member is K‑semistable | |||||||||
| 3-11 | blowup of 2-35 in an elliptic curve which is the intersection of two divisors from $|-\frac{1}{2}\mathrm{K}_{V_7}|$ | 2 | $0$ | general member is K‑stable | general member is K‑polystable | general member is K‑semistable | |||||||||
| 3-12 | blowup of 1-17 in the disjoint union of a line and a twisted cubic | 1 |
| general member is K‑stable but there exist members that are not | general member is K‑polystable but there exist members that are not | every member is K‑semistable | |||||||||
| 3-13 | blowup of 2-32 in a curve $C$ of bidegree $(2,2)$ such that the composition $C\hookrightarrow W\hookrightarrow\mathbb{P}^2\times\mathbb{P}^2\overset{p_i}{\to}\mathbb{P}^2$ is an embedding for $i=1,2$ | 1 |
| none are K-stable | general member is K‑polystable but there exist members that are not | every member is K‑semistable | |||||||||
| 3-14 | blowup of 1-17 in the disjoint union of a plane cubic curve and a point outside the plane | 1 | $\mathbb{G}_{\mathrm{m}}$ | none are K-stable | none are K-polystable | none are K-semistable | |||||||||
| 3-15 | blowup of 1-16 in the disjoint union of a line and a conic | 0 | $\mathbb{G}_{\mathrm{m}}$ | none are K-stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 3-16 | blowup of 2-35 in the proper transform of a twisted cubic containing the center of the blowup | 0 | $\mathrm{B}$ | none are K-stable | none are K-polystable | none are K-semistable | |||||||||
| 3-17 | divisor on $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^2$ of degree $(1,1,1)$ | 0 | $\mathrm{PGL}_2$ | none are K-stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 3-18 | blowup of 1-17 in the disjoint union of a line and a conic | 0 | $\mathrm{B}\times\mathbb{G}_{\mathrm{m}}$ | none are K-stable | none are K-polystable | none are K-semistable | |||||||||
| 3-19 | blowup of 1-16 in two non-collinear points | 0 | $\mathbb{G}_{\mathrm{m}}\times\mathrm{PGL}_2$ | none are K-stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 3-20 | blowup of 1-16 in the disjoint union of two lines | 0 | $\mathbb{G}_{\mathrm{m}}\times\mathrm{PGL}_2$ | none are K-stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 3-21 | blowup of 2-34 in a curve of degree $(2,1)$ | 0 | $\mathbb{G}_{\mathrm{a}}^2\rtimes\mathbb{G}_{\mathrm{m}}^2$ | none are K-stable | none are K-polystable | none are K-semistable | |||||||||
| 3-22 | blowup of 2-34 in a conic on $\{x\}\times\mathbb{P}^2$, $x\in\mathbb{P}^1$ | 0 | $\mathrm{B}\times\mathrm{PGL}_2$ | none are K-stable | none are K-polystable | none are K-semistable | |||||||||
| 3-23 | blowup of 2-35 in the proper transform of a conic containing the center of the blowup | 0 | $\mathbb{G}_{\mathrm{a}}^3\rtimes(\mathrm{B}\times\mathbb{G}_{\mathrm{m}})$ | none are K-stable | none are K-polystable | none are K-semistable | |||||||||
| 3-24 | the fiber product of 2-32 with $\mathrm{Bl}_p\mathbb{P}^2$ over $\mathbb{P}^2$
| 0 | $\mathrm{PGL}_{3;1}$ | none are K-stable | none are K-polystable | none are K-semistable | |||||||||
| 3-25 | blowup of 1-17 in the disjoint union of two lines
| 0 | $\mathrm{PGL}_{(2,2)}$ | none are K-stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 3-26 | blowup of 1-17 in the disjoint union of a point and a line
| 0 | $\mathbb{G}_{\mathrm{a}}^3\rtimes(\mathrm{GL}_2\times\mathbb{G}_{\mathrm{m}})$ | none are K-stable | none are K-polystable | none are K-semistable | |||||||||
| 3-27 | $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ | 0 | $\mathrm{PGL}_2^3$ | none are K-stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 3-28 | $\mathbb{P}^1\times\mathrm{Bl}_p\mathbb{P}^2$ | 0 | $\mathrm{PGL}_2\times\mathrm{PGL}_{3;1}$ | none are K-stable | none are K-polystable | none are K-semistable | |||||||||
| 3-29 | blowup of 2-35 in a line on the exceptional divisor | 0 | $\mathrm{PGL}_{4;3,1}$ | none are K-stable | none are K-polystable | none are K-semistable | |||||||||
| 3-30 | blowup of 2-35 in the proper transform of a line containing the center of the blowup
| 0 | $\mathrm{PGL}_{4;2,1}$ | none are K-stable | none are K-polystable | none are K-semistable | |||||||||
| 3-31 | blowup of the cone over a smooth quadric in $\mathbb{P}^3$ in the vertex
| 0 | $\mathrm{PSO}_{6;1}$ | none are K-stable | none are K-polystable | none are K-semistable | |||||||||
| 4-1 | divisor on $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ of degree $(1,1,1,1)$ | 3 | $0$ | every member is K‑stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 4-2 | blowup of the cone over a smooth quadric in $\mathbb{P}^3$ in the disjoint union of the vertex and an elliptic curve on the quadric | 2 | $\mathbb{G}_{\mathrm{m}}$ | none are K-stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 4-3 | blowup of 3-27 in a curve of degree $(1,1,2)$ | 0 | $\mathbb{G}_{\mathrm{m}}$ | none are K-stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 4-4 | blowup of 3-19 in the proper transform of a conic through the points | 0 | $\mathbb{G}_{\mathrm{m}}^2$ | none are K-stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 4-5 | blowup of 2-34 in the disjoint union of a curve of degree $(2,1)$ and a curve of degree $(1,0)$ | 0 | $\mathbb{G}_{\mathrm{m}}^2$ | none are K-stable | none are K-polystable | none are K-semistable | |||||||||
| 4-6 | blowup of 1-17 in the disjoint union of 3 lines
| 0 | $\mathrm{PGL}_2$ | none are K-stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 4-7 | blowup of 2-32 in the disjoint union of a curve of degree $(0,1)$ and a curve of degree $(1,0)$ | 0 | $\mathrm{GL}_2$ | none are K-stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 4-8 | blowup of 3-27 in a curve of degree $(0,1,1)$ | 0 | $\mathrm{B}\times\mathrm{PGL}_2$ | none are K-stable | none are K-polystable | none are K-semistable | |||||||||
| 4-9 | blowup of 3-25 in an exceptional curve of the blowup | 0 | $\mathrm{PGL}_{(2,2);1}$ | none are K-stable | none are K-polystable | none are K-semistable | |||||||||
| 4-10 | $\mathbb{P}^1\times\mathrm{Bl}_2\mathbb{P}^2$ | 0 | $\mathrm{PGL}_2\times\mathrm{B}^2$ | none are K-stable | none are K-polystable | none are K-semistable | |||||||||
| 4-11 | blowup of 3-28 in $\{x\}\times E$, $x\in\mathbb{P}^1$ and $E$ the $(-1)$-curve | 0 | $\mathrm{B}\times\mathrm{PGL}_{3;1}$ | none are K-stable | none are K-polystable | none are K-semistable | |||||||||
| 4-12 | blowup of 2-33 in the disjoint union of two exceptional lines of the blowup | 0 | $\mathbb{G}_{\mathrm{a}}^4\rtimes(\mathrm{GL}_2\times\mathbb{G}_{\mathrm{m}})$ | none are K-stable | none are K-polystable | none are K-semistable | |||||||||
| 4-13 | blowup of 3-27 in a curve of degree $(1,1,3)$ | 1 |
| general member is K‑stable but there exist members that are not | general member is K‑polystable but there exist members that are not | every member is K‑semistable | |||||||||
| 5-1 | blowup of 2-29 in the disjoint union of three exceptional lines of the blowup | 0 | $\mathbb{G}_{\mathrm{m}}$ | none are K-stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 5-2 | blowup of 3-25 in the disjoint union of two exceptional lines on the same irreducible component | 0 | $\mathbb{G}_{\mathrm{m}}\times\mathrm{GL}_2$ | none are K-stable | none are K-polystable | none are K-semistable | |||||||||
| 5-3 | $\mathbb{P}^1\times\mathrm{Bl}_3\mathbb{P}^2$ | 0 | $\mathrm{PGL}_2\times\mathbb{G}_{\mathrm{m}}^2$ | none are K-stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 6-1 | $\mathbb{P}^1\times\mathrm{Bl}_4\mathbb{P}^2$ | 0 | $\mathrm{PGL}_2$ | none are K-stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 7-1 | $\mathbb{P}^1\times\mathrm{Bl}_5\mathbb{P}^2$ | 2 | $\mathrm{PGL}_2$ | none are K-stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 8-1 | $\mathbb{P}^1\times\mathrm{Bl}_6\mathbb{P}^2$ | 4 | $\mathrm{PGL}_2$ | none are K-stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 9-1 | $\mathbb{P}^1\times\mathrm{Bl}_7\mathbb{P}^2$ | 6 | $\mathrm{PGL}_2$ | none are K-stable | every member is K‑polystable | every member is K‑semistable | |||||||||
| 10-1 | $\mathbb{P}^1\times\mathrm{Bl}_8\mathbb{P}^2$ | 8 | $\mathrm{PGL}_2$ | none are K-stable | every member is K‑polystable | every member is K‑semistable |
Open cases
K-(poly/semi)stability is not fully understood for (the smooth members in) the following families: