Fanography

A tool to visually study the geography of Fano 3-folds.

This is currently Table 6.1 of the book The Calabi problem for Fano threefolds, so more recent progress is missing at the moment.

Below the table we provide an overview of the open cases.

ID description moduli $\mathrm{Aut}^0$ K‑stability K‑polystability K‑semistability
1-1

double cover of $\mathbb{P}^3$ with branch locus a divisor of degree 6

alternative
hypersurface of degree 6 in $\mathbb{P}(1,1,1,1,3)$

68

$0$

K-stability is not understood every member is K-polystable every member is K-semistable
1-2
  1. hypersurface of degree 4 in $\mathbb{P}^4$
  2. double cover of 1-16 with branch locus a divisor of degree 8
  1. 45
  2. 44

$0$

K-stability is not understood every member is K-polystable every member is K-semistable
1-3

complete intersection of quadric and cubic in $\mathbb{P}^5$

34

$0$

K-stability is not understood every member is K-polystable every member is K-semistable
1-4

complete intersection of 3 quadrics in $\mathbb{P}^6$

27

$0$

K-stability is not understood every member is K-polystable every member is K-semistable
1-5 Gushel–Mukai 3-fold
  1. section of Plücker embedding of $\mathrm{Gr}(2,5)$ by codimension 2 subspace and a quadric
  2. double cover of 1-15 with branch locus an anticanonical divisor
  1. 22
  2. 19

$0$

K-stability is not understood 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

$0$

K-stability is not understood 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

$0$

K-stability is not understood 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

$0$

K-stability is not understood 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

$0$

K-stability is not understood 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

$\mathrm{Aut}^0(X)$ moduli
$\mathrm{PGL}_2$ 0
$\mathbb{G}_{\mathrm{a}}$ 0
$\mathbb{G}_{\mathrm{m}}$ 1
K-stability is not understood general member is K-polystable but there exists one that is not general member is K-semistable
1-11

hypersurface of degree 6 in $\mathbb{P}(1,1,1,2,3)$

34

$0$

K-stability is not understood every member is K-polystable every member is K-semistable
1-12

double cover of $\mathbb{P}^3$ with branch locus a smooth quartic surface

alternative
hypersurface of degree 4 in $\mathbb{P}(1,1,1,1,2)$

19

$0$

K-stability is not understood every member is K-polystable every member is K-semistable
1-13

hypersurface of degree 3 in $\mathbb{P}^4$

10

$0$

K-stability is not understood every member is K-polystable every member is K-semistable
1-14

complete intersection of 2 quadrics in $\mathbb{P}^5$

3

$0$

K-stability is not understood every member is K-polystable every member is K-semistable
1-15

section of Plücker embedding of $\mathrm{Gr}(2,5)$ by codimension 3 subspace

0

$\mathrm{PGL}_2$

K-stability is not understood every member is K-polystable every member is K-semistable
1-16

hypersurface of degree 2 in $\mathbb{P}^4$

0

$\mathrm{PSO}_5$

K-stability is not understood every member is K-polystable every member is K-semistable
1-17

projective space $\mathbb{P}^3$

0

$\mathrm{PGL}_4$

K-stability is not understood 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

alternative
$(1,3)$-divisor on $\mathbb{P}^1\times\mathbb{P}^3$

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
  1. $(2,2)$-divisor on $\mathbb{P}^2\times\mathbb{P}^2$
  2. double cover of 2-32 with branch locus an anticanonical divisor
  1. 19
  2. 18

$0$

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
  1. double cover of 2-35 with branch locus an anticanonical divisor such that the intersection with the exceptional divisor is smooth
  2. double cover of 2-35 with branch locus an anticanonical divisor such that the intersection with the exceptional divisor is singular but reduced
  1. 18
  2. 17

$0$

K-stability is not understood every member is K-polystable every member is K-semistable
2-9

complete intersection of degree $(1,1)$ and $(2,1)$ in $\mathbb{P}^3\times\mathbb{P}^2$

alternative
blowup of 1-17 in a curve of degree 7 and genus 5, which is an intersection of 3 cubics

13

$0$

K-stability is not understood 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

$0$

K-stability is not understood general member is K-polystable general member is K-semistable
2-11

blowup of 1-13 in a line

12

$0$

K-stability is not understood general member is K-polystable general member is K-semistable
2-12

intersection of 3 $(1,1)$-divisors in $\mathbb{P}^3\times\mathbb{P}^3$

alternative
blowup of 1-17 in a curve of degree 6 and genus 3 which is an intersection of 4 cubics

9

$0$

K-stability is not understood 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

$0$

K-stability is not understood 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

$0$

K-stability is not understood general member is K-polystable general member is K-semistable
2-15
  1. blowup of 1-17 in the intersection of a quadric and a cubic where the quadric is smooth
  2. blowup of 1-17 in the intersection of a quadric and a cubic where the quadric is singular but reduced
  1. 9
  2. 8

$0$

K-stability is not understood every member is K-polystable every member is K-semistable
2-16

blowup of 1-14 in a conic

7

$0$

K-stability is not understood general member is K-polystable general member is K-semistable
2-17

blowup of 1-16 in an elliptic curve of degree 5

5

$0$

K-stability is not understood 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

$0$

K-stability is not understood general member is K-polystable general member is K-semistable
2-20

blowup of 1-15 in a twisted cubic

3

$\mathrm{Aut}^0(X)$ moduli
$\mathbb{G}_{\mathrm{m}}$ 0
K-stability is not understood general member is K-polystable but there exists one that is not general member is K-semistable
2-21

blowup of 1-16 in a twisted quartic

2

$\mathrm{Aut}^0(X)$ moduli
$\mathrm{PGL}_2$ 0
$\mathbb{G}_{\mathrm{a}}$ 0
$\mathbb{G}_{\mathrm{m}}$ 1
K-stability is not understood general member is K-polystable but there exists one that is not general member is K-semistable
2-22

blowup of 1-15 in a conic

1

$\mathrm{Aut}^0(X)$ moduli
$\mathbb{G}_{\mathrm{m}}$ 0
general member is K-stable but there exists one that is not general member is K-polystable but there exists one that is not every member is K-semistable
2-23
  1. blowup of 1-16 in an intersection of $A\in|\mathcal{O}_Q(1)|$ and $B\in|\mathcal{O}_Q(2)|$ such that $A$ is smooth
  2. blowup of 1-16 in an intersection of $A\in|\mathcal{O}_Q(1)|$ and $B\in|\mathcal{O}_Q(2)|$ such that $A$ is singular
  1. 2
  2. 1

$0$

K-stability is not understood none are K-polystable none are K-semistable
2-24

divisor on $\mathbb{P}^2\times\mathbb{P}^2$ of bidegree $(1,2)$

1

$\mathrm{Aut}^0(X)$ moduli
$\mathbb{G}_{\mathrm{m}}^2$ 0
$\mathbb{G}_{\mathrm{m}}$ 0
K-stability is not understood general member is K-polystable but there exists one that is not general member is K-semistable
2-25

blowup of 1-17 in an elliptic curve which is an intersection of 2 quadrics

alternative
$(1,2)$-divisor on $\mathbb{P}^1\times\mathbb{P}^3$

1

$0$

K-stability is not understood every member is K-polystable every member is K-semistable
2-26

blowup of 1-15 in a line

0

$\mathrm{Aut}^0(X)$ moduli
$\mathrm{B}$ 0
$\mathbb{G}_{\mathrm{m}}$ 0
K-stability is not understood none are K-polystable general member is K-semistable but there exists one that is not
2-27

blowup of 1-17 in a twisted cubic

0

$\mathrm{PGL}_2$

K-stability is not understood every member is K-polystable every member is K-semistable
2-28

blowup of 1-17 in a plane cubic

1

$\mathbb{G}_{\mathrm{a}}^3\rtimes\mathbb{G}_{\mathrm{m}}$

K-stability is not understood none are K-polystable none are K-semistable
2-29

blowup of 1-16 in a conic

0

$\mathbb{G}_{\mathrm{m}}\times\mathrm{PGL}_2$

K-stability is not understood every member is K-polystable every member is K-semistable
2-30

blowup of 1-17 in a conic

0

$\mathrm{PSO}_{5;1}$

K-stability is not understood none are K-polystable none are K-semistable
2-31

blowup of 1-16 in a line

0

$\mathrm{PSO}_{5;2}$

K-stability is not understood none are K-polystable none are K-semistable
2-32

divisor on $\mathbb{P}^2\times\mathbb{P}^2$ of bidegree $(1,1)$

alternative
$\mathbb{P}(\mathrm{T}_{\mathbb{P}^2})$
the complete flag variety for $\mathbb{P}^2$

0

$\mathrm{PGL}_3$

K-stability is not understood every member is K-polystable every member is K-semistable
2-33

blowup of 1-17 in a line

0

$\mathrm{PGL}_{4;2}$

K-stability is not understood none are K-polystable none are K-semistable
2-34

$\mathbb{P}^1\times\mathbb{P}^2$

0

$\mathrm{PGL}_2\times\mathrm{PGL}_3$

K-stability is not understood every member is K-polystable every member is K-semistable
2-35

$\mathrm{Bl}_p\mathbb{P}^3$

alternative
$\mathbb{P}(\mathcal{O}_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}(1))$

0

$\mathrm{PGL}_{4;1}$

K-stability is not understood 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

$\mathrm{Aut}(\mathbb{P}(1,1,1,2))$

K-stability is not understood 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

$0$

K-stability is not understood 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

$0$

K-stability is not understood 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

$\mathrm{Aut}^0(X)$ moduli
$\mathbb{G}_{\mathrm{m}}$ 0
general member is K-stable but there exists one that is not general member is K-polystable but there exists one that is 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

alternative
complete intersection of degree $(1,0,2)$ and $(0,1,1)$ in $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^3$

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

$0$

K-stability is not understood 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

$\mathrm{Aut}^0(X)$ moduli
$\mathbb{G}_{\mathrm{m}}$ 0
K-stability is not understood general member is K-polystable but there exists one that is 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

$\mathbb{G}_{\mathrm{m}}$

K-stability is not understood every member is K-polystable every member is K-semistable
3-10

blowup of 1-16 in the disjoint union of 2 conics

alternative
complete intersection of degree $(1,0,1)$, $(0,1,1)$ and $(0,0,2)$ in $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^4$

2

$\mathrm{Aut}^0(X)$ moduli
$\mathbb{G}_{\mathrm{m}}^2$ 0
$\mathbb{G}_{\mathrm{m}}$ 1
K-stability is not understood general member is K-polystable but there exists one that is 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

$0$

K-stability is not understood 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

$\mathrm{Aut}^0(X)$ moduli
$\mathbb{G}_{\mathrm{m}}$ 0
K-stability is not understood general member is K-polystable but there exists one that is 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

$\mathrm{Aut}^0(X)$ moduli
$\mathrm{PGL}_2$ 0
$\mathbb{G}_{\mathrm{a}}$ 0
$\mathbb{G}_{\mathrm{m}}$ 1
K-stability is not understood general member is K-polystable but there exists one that is 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

$\mathbb{G}_{\mathrm{m}}$

K-stability is not understood 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

$\mathbb{G}_{\mathrm{m}}$

K-stability is not understood 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

$\mathrm{B}$

K-stability is not understood 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

$\mathrm{PGL}_2$

K-stability is not understood 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

$\mathrm{B}\times\mathbb{G}_{\mathrm{m}}$

K-stability is not understood none are K-polystable none are K-semistable
3-19

blowup of 1-16 in two non-collinear points

0

$\mathbb{G}_{\mathrm{m}}\times\mathrm{PGL}_2$

K-stability is not understood every member is K-polystable every member is K-semistable
3-20

blowup of 1-16 in the disjoint union of two lines

0

$\mathbb{G}_{\mathrm{m}}\times\mathrm{PGL}_2$

K-stability is not understood every member is K-polystable every member is K-semistable
3-21

blowup of 2-34 in a curve of degree $(2,1)$

0

$\mathbb{G}_{\mathrm{a}}^2\rtimes\mathbb{G}_{\mathrm{m}}^2$

K-stability is not understood 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

$\mathrm{B}\times\mathrm{PGL}_2$

K-stability is not understood 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

$\mathbb{G}_{\mathrm{a}}^3\rtimes(\mathrm{B}\times\mathbb{G}_{\mathrm{m}})$

K-stability is not understood none are K-polystable none are K-semistable
3-24

the fiber product of 2-32 with $\mathrm{Bl}_p\mathbb{P}^2$ over $\mathbb{P}^2$

alternative
complete intersection of degree $(1,1,0)$ and $(0,1,1)$ in $\mathbb{P}^1\times\mathbb{P}^2\times\mathbb{P}^2$

0

$\mathrm{PGL}_{3;1}$

K-stability is not understood none are K-polystable none are K-semistable
3-25

blowup of 1-17 in the disjoint union of two lines

alternative
$\mathbb{P}(\mathcal{O}(1,0)\oplus\mathcal{O}(0,1))$ over $\mathbb{P}^1\times\mathbb{P}^1$

0

$\mathrm{PGL}_{(2,2)}$

K-stability is not understood 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

alternative
blowup of line on a plane which is section of 2-34 mapping to $\mathbb{P}^2$

0

$\mathbb{G}_{\mathrm{a}}^3\rtimes(\mathrm{GL}_2\times\mathbb{G}_{\mathrm{m}})$

K-stability is not understood none are K-polystable none are K-semistable
3-27

$\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$

0

$\mathrm{PGL}_2^3$

K-stability is not understood every member is K-polystable every member is K-semistable
3-28

$\mathbb{P}^1\times\mathrm{Bl}_p\mathbb{P}^2$

0

$\mathrm{PGL}_2\times\mathrm{PGL}_{3;1}$

K-stability is not understood none are K-polystable none are K-semistable
3-29

blowup of 2-35 in a line on the exceptional divisor

0

$\mathrm{PGL}_{4;3,1}$

K-stability is not understood 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

alternative
$\mathbb{P}_{\mathbb{F}_1}(\mathcal{O}\oplus\mathcal{O}(\ell))$ where $\ell^2=1$

0

$\mathrm{PGL}_{4;2,1}$

K-stability is not understood 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

alternative
$\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(1,1))$ over $\mathbb{P}^1\times\mathbb{P}^1$

0

$\mathrm{PSO}_{6;1}$

K-stability is not understood 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

$0$

K-stability is not understood 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

$\mathbb{G}_{\mathrm{m}}$

K-stability is not understood 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

$\mathbb{G}_{\mathrm{m}}$

K-stability is not understood 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

$\mathbb{G}_{\mathrm{m}}^2$

K-stability is not understood 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

$\mathbb{G}_{\mathrm{m}}^2$

K-stability is not understood none are K-polystable none are K-semistable
4-6

blowup of 1-17 in the disjoint union of 3 lines

alternative
blowup of 3-27 in the tridiagonal

0

$\mathrm{PGL}_2$

K-stability is not understood 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

$\mathrm{GL}_2$

K-stability is not understood 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

$\mathrm{B}\times\mathrm{PGL}_2$

K-stability is not understood none are K-polystable none are K-semistable
4-9

blowup of 3-25 in an exceptional curve of the blowup

0

$\mathrm{PGL}_{(2,2);1}$

K-stability is not understood none are K-polystable none are K-semistable
4-10

$\mathbb{P}^1\times\mathrm{Bl}_2\mathbb{P}^2$

0

$\mathrm{PGL}_2\times\mathrm{B}^2$

K-stability is not understood 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

$\mathrm{B}\times\mathrm{PGL}_{3;1}$

K-stability is not understood 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

$\mathbb{G}_{\mathrm{a}}^4\rtimes(\mathrm{GL}_2\times\mathbb{G}_{\mathrm{m}})$

K-stability is not understood none are K-polystable none are K-semistable
4-13

blowup of 3-27 in a curve of degree $(1,1,3)$

1

$\mathrm{Aut}^0(X)$ moduli
$\mathbb{G}_{\mathrm{m}}$ 0
K-stability is not understood general member is K-polystable but there exists one that is not general member is K-semistable
5-1

blowup of 2-29 in the disjoint union of three exceptional lines of the blowup

0

$\mathbb{G}_{\mathrm{m}}$

K-stability is not understood 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

$\mathbb{G}_{\mathrm{m}}\times\mathrm{GL}_2$

K-stability is not understood none are K-polystable none are K-semistable
5-3

$\mathbb{P}^1\times\mathrm{Bl}_3\mathbb{P}^2$

0

$\mathrm{PGL}_2\times\mathbb{G}_{\mathrm{m}}^2$

K-stability is not understood every member is K-polystable every member is K-semistable
6-1

$\mathbb{P}^1\times\mathrm{Bl}_4\mathbb{P}^2$

0

$\mathrm{PGL}_2$

K-stability is not understood every member is K-polystable every member is K-semistable
7-1

$\mathbb{P}^1\times\mathrm{Bl}_5\mathbb{P}^2$

2

$\mathrm{PGL}_2$

K-stability is not understood every member is K-polystable every member is K-semistable
8-1

$\mathbb{P}^1\times\mathrm{Bl}_6\mathbb{P}^2$

4

$\mathrm{PGL}_2$

K-stability is not understood every member is K-polystable every member is K-semistable
9-1

$\mathbb{P}^1\times\mathrm{Bl}_7\mathbb{P}^2$

6

$\mathrm{PGL}_2$

K-stability is not understood every member is K-polystable every member is K-semistable
10-1

$\mathbb{P}^1\times\mathrm{Bl}_8\mathbb{P}^2$

8

$\mathrm{PGL}_2$

K-stability is not understood every member is K-polystable every member is K-semistable

Open cases

K-stability is not fully understood for the following families:

K-polystability is not fully understood for the following families:

K-semistability is not fully understood for the following families: