Fano threefolds with $\rho=1$
ID | $-\mathrm{K}_X^3$ | $g$ | $\mathrm{h}^{1,2}$ | index | description | blowups | rational | unirational | moduli | $\mathrm{Aut}^0$ | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1-1 | 2 | 2 | 52 | 1 |
double cover of $\mathbb{P}^3$ with branch locus a divisor of degree 6
|
no | ? | 68 | $0$ | |||||||||
1-2 | 4 | 3 | 30 | 1 |
|
no | some |
|
$0$ | |||||||||
1-3 | 6 | 4 | 20 | 1 |
complete intersection of quadric and cubic in $\mathbb{P}^5$ |
no | yes | 34 | $0$ | |||||||||
1-4 | 8 | 5 | 14 | 1 |
complete intersection of 3 quadrics in $\mathbb{P}^6$ |
no | yes | 27 | $0$ | |||||||||
1-5 | 10 | 6 | 10 | 1 |
Gushel–Mukai 3-fold
|
generically non-rational | yes |
|
$0$ | |||||||||
1-6 | 12 | 7 | 7 | 1 |
section of half-spinor embedding of a connected component of $\mathrm{OGr}_+(5,10)$ by codimension 7 subspace |
yes | yes | 18 | $0$ | |||||||||
1-7 | 14 | 8 | 5 | 1 |
section of Plücker embedding of $\mathrm{Gr}(2,6)$ by codimension 5 subspace |
no | yes | 15 | $0$ | |||||||||
1-8 | 16 | 9 | 3 | 1 |
section of Plücker embedding of $\mathrm{SGr}(3,6)$ by codimension 3 subspace |
yes | yes | 12 | $0$ | |||||||||
1-9 | 18 | 10 | 2 | 1 |
section of the adjoint $\mathrm{G}_2$-Grassmannian $\mathrm{G}_2\mathrm{Gr}(2,7)$ by codimension 2 subspace |
yes | yes | 10 | $0$ | |||||||||
1-10 | 22 | 12 | 0 | 1 |
zero locus of $(\bigwedge^2\mathcal{U}^\vee)^{\oplus 3}$ on $\mathrm{Gr}(3,7)$ |
yes | yes | 6 |
|
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1-11 | 8 | 21 | 2 |
hypersurface of degree 6 in $\mathbb{P}(1,1,1,2,3)$ |
* | no | ? | 34 | $0$ | |||||||||
1-12 | 16 | 10 | 2 |
double cover of $\mathbb{P}^3$ with branch locus a smooth quartic surface
|
* | no | yes | 19 | $0$ | |||||||||
1-13 | 24 | 5 | 2 |
hypersurface of degree 3 in $\mathbb{P}^4$ |
* | no | yes | 10 | $0$ | |||||||||
1-14 | 32 | 2 | 2 |
complete intersection of 2 quadrics in $\mathbb{P}^5$ |
* | yes | yes | 3 | $0$ | |||||||||
1-15 | 40 | 0 | 2 |
section of Plücker embedding of $\mathrm{Gr}(2,5)$ by codimension 3 subspace |
* | yes | yes | 0 | $\mathrm{PGL}_2$ | |||||||||
1-16 | 54 | 0 | 3 |
hypersurface of degree 2 in $\mathbb{P}^4$ |
* | yes | yes | 0 | $\mathrm{PSO}_5$ | |||||||||
1-17 | 64 | 0 | 4 |
projective space $\mathbb{P}^3$ |
* | yes | yes | 0 | $\mathrm{PGL}_4$ |
Fano threefolds with $\rho=2$
ID | $-\mathrm{K}_X^3$ | $\mathrm{h}^{1,2}$ | index | description | blowups | blowdowns | rational | unirational | moduli | $\mathrm{Aut}^0$ | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2-1 | 4 | 22 | 1 |
blowup of 1-11 in an elliptic curve which is the intersection of two divisors from half the anticanonical linear system |
1-11 | no | ? | 36 | $0$ | |||||||||
2-2 | 6 | 20 | 1 |
double cover of 2-34 with branch locus a $(2,4)$-divisor |
no | yes | 33 | $0$ | ||||||||||
2-3 | 8 | 11 | 1 |
blowup of 1-12 in an elliptic curve which is the intersection of two divisors from half the anticanonical linear system |
1-12 | no | yes | 23 | $0$ | |||||||||
2-4 | 10 | 10 | 1 |
blowup of 1-17 in the intersection of two cubics
|
1-17 | yes | yes | 21 | $0$ | |||||||||
2-5 | 12 | 6 | 1 |
blowup of 1-13 in a plane cubic |
1-13 | no | yes | 16 | $0$ | |||||||||
2-6 | 12 | 9 | 1 |
Verra 3-fold
|
no | yes |
|
$0$ | ||||||||||
2-7 | 14 | 5 | 1 |
blowup of 1-16 in the intersection of two divisors from $|\mathcal{O}_Q(2)|$ |
1-16 | yes | yes | 14 | $0$ | |||||||||
2-8 | 14 | 9 | 1 | no | yes |
|
$0$ | |||||||||||
2-9 | 16 | 5 | 1 |
complete intersection of degree $(1,1)$ and $(2,1)$ in $\mathbb{P}^3\times\mathbb{P}^2$
|
1-17 | yes | yes | 13 | $0$ | |||||||||
2-10 | 16 | 3 | 1 |
blowup of 1-14 in an elliptic curve which is an intersection of 2 hyperplanes |
1-14 | yes | yes | 11 | $0$ | |||||||||
2-11 | 18 | 5 | 1 |
blowup of 1-13 in a line |
1-13 | no | yes | 12 | $0$ | |||||||||
2-12 | 20 | 3 | 1 |
intersection of 3 $(1,1)$-divisors in $\mathbb{P}^3\times\mathbb{P}^3$
|
1-17 | yes | yes | 9 | $0$ | |||||||||
2-13 | 20 | 2 | 1 |
blowup of 1-16 in a curve of degree 6 and genus 2 |
1-16 | yes | yes | 8 | $0$ | |||||||||
2-14 | 20 | 1 | 1 |
blowup of 1-15 in an elliptic curve which is an intersection of 2 hyperplanes |
1-15 | yes | yes | 7 | $0$ | |||||||||
2-15 | 22 | 4 | 1 | 1-17 | yes | yes |
|
$0$ | ||||||||||
2-16 | 22 | 2 | 1 |
blowup of 1-14 in a conic |
1-14 | yes | yes | 7 | $0$ | |||||||||
2-17 | 24 | 1 | 1 |
blowup of 1-16 in an elliptic curve of degree 5 |
1-16, 1-17 | yes | yes | 5 | $0$ | |||||||||
2-18 | 24 | 2 | 1 |
double cover of 2-34 with branch locus a divisor of degree $(2,2)$ |
* | yes | yes | 6 | $0$ | |||||||||
2-19 | 26 | 2 | 1 |
blowup of 1-14 in a line |
1-14, 1-17 | yes | yes | 5 | $0$ | |||||||||
2-20 | 26 | 0 | 1 |
blowup of 1-15 in a twisted cubic |
1-15 | yes | yes | 3 |
|
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2-21 | 28 | 0 | 1 |
blowup of 1-16 in a twisted quartic |
1-16 | yes | yes | 2 |
|
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2-22 | 30 | 0 | 1 |
blowup of 1-15 in a conic |
1-15, 1-17 | yes | yes | 1 |
|
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2-23 | 30 | 1 | 1 | 1-16 | yes | yes |
|
$0$ | ||||||||||
2-24 | 30 | 0 | 1 |
divisor on $\mathbb{P}^2\times\mathbb{P}^2$ of bidegree $(1,2)$ |
* | yes | yes | 1 |
|
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2-25 | 32 | 1 | 1 |
blowup of 1-17 in an elliptic curve which is an intersection of 2 quadrics
|
* | 1-17 | yes | yes | 1 | $0$ | ||||||||
2-26 | 34 | 0 | 1 |
blowup of 1-15 in a line |
1-15, 1-16 | yes | yes | 0 |
|
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2-27 | 38 | 0 | 1 |
blowup of 1-17 in a twisted cubic |
* | 1-17 | yes | yes | 0 | $\mathrm{PGL}_2$ | ||||||||
2-28 | 40 | 1 | 1 |
blowup of 1-17 in a plane cubic |
1-17 | yes | yes | 1 | $\mathbb{G}_{\mathrm{a}}^3\rtimes\mathbb{G}_{\mathrm{m}}$ | |||||||||
2-29 | 40 | 0 | 1 |
blowup of 1-16 in a conic |
* | 1-16 | yes | yes | 0 | $\mathbb{G}_{\mathrm{m}}\times\mathrm{PGL}_2$ | ||||||||
2-30 | 46 | 0 | 1 |
blowup of 1-17 in a conic |
* | 1-17 | yes | yes | 0 | $\mathrm{PSO}_{5;1}$ | ||||||||
2-31 | 46 | 0 | 1 |
blowup of 1-16 in a line |
* | 1-16 | yes | yes | 0 | $\mathrm{PSO}_{5;2}$ | ||||||||
2-32 | 48 | 0 | 2 |
divisor on $\mathbb{P}^2\times\mathbb{P}^2$ of bidegree $(1,1)$
|
* | yes | yes | 0 | $\mathrm{PGL}_3$ | |||||||||
2-33 | 54 | 0 | 1 |
blowup of 1-17 in a line |
* | 1-17 | yes | yes | 0 | $\mathrm{PGL}_{4;2}$ | ||||||||
2-34 | 54 | 0 | 1 |
$\mathbb{P}^1\times\mathbb{P}^2$ |
* | yes | yes | 0 | $\mathrm{PGL}_2\times\mathrm{PGL}_3$ | |||||||||
2-35 | 56 | 0 | 2 |
$\mathrm{Bl}_p\mathbb{P}^3$
|
* | yes | yes | 0 | $\mathrm{PGL}_{4;1}$ | |||||||||
2-36 | 62 | 0 | 1 |
$\mathbb{P}(\mathcal{O}_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}(2))$ |
* | yes | yes | 0 | $\mathrm{Aut}(\mathbb{P}(1,1,1,2))$ |
Fano threefolds with $\rho=3$
ID | $-\mathrm{K}_X^3$ | $\mathrm{h}^{1,2}$ | index | description | blowups | blowdowns | rational | unirational | moduli | $\mathrm{Aut}^0$ | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3-1 | 12 | 8 | 1 |
double cover of 3-27 with branch locus a divisor of degree $(2,2,2)$ |
no | yes | 17 | $0$ | ||||||||||
3-2 | 14 | 3 | 1 |
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 $Y\in|\mathcal{L}|$ |
yes | yes | 11 | $0$ | ||||||||||
3-3 | 18 | 3 | 1 |
divisor on $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^2$ of degree $(1,1,2)$ |
2-34 | yes | yes | 9 | $0$ | |||||||||
3-4 | 18 | 2 | 1 |
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 |
2-18 | yes | yes | 8 | $0$ | |||||||||
3-5 | 20 | 0 | 1 |
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 |
2-34 | yes | yes | 5 |
|
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3-6 | 22 | 1 | 1 |
blowup of 1-17 in the disjoint union of a line and an elliptic curve of degree 4
|
2-25, 2-33 | yes | yes | 5 | $0$ | |||||||||
3-7 | 24 | 1 | 1 |
blowup of 2-32 in an elliptic curve which is the intersection of two divisors from $|-\frac{1}{2}\mathrm{K}_W|$ |
2-32, 2-34 | yes | yes | 4 | $0$ | |||||||||
3-8 | 24 | 0 | 1 |
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 |
2-24, 2-34 | yes | yes | 3 |
|
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3-9 | 26 | 3 | 1 |
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 $\mathbb{P}^2$ |
2-36 | yes | yes | 6 | $\mathbb{G}_{\mathrm{m}}$ | |||||||||
3-10 | 26 | 0 | 1 |
blowup of 1-16 in the disjoint union of 2 conics
|
2-29 | yes | yes | 2 |
|
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3-11 | 28 | 1 | 1 |
blowup of 2-35 in an elliptic curve which is the intersection of two divisors from $|-\frac{1}{2}\mathrm{K}_{V_7}|$ |
2-25, 2-34, 2-35 | yes | yes | 2 | $0$ | |||||||||
3-12 | 28 | 0 | 1 |
blowup of 1-17 in the disjoint union of a line and a twisted cubic |
2-27, 2-33, 2-34 | yes | yes | 1 |
|
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3-13 | 30 | 0 | 1 |
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 $i=1,2$ |
2-32 | yes | yes | 1 |
|
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3-14 | 32 | 1 | 1 |
blowup of 1-17 in the disjoint union of a plane cubic curve and a point outside the plane |
2-35, 2-36 | yes | yes | 1 | $\mathbb{G}_{\mathrm{m}}$ | |||||||||
3-15 | 32 | 0 | 1 |
blowup of 1-16 in the disjoint union of a line and a conic |
2-29, 2-31, 2-34 | yes | yes | 0 | $\mathbb{G}_{\mathrm{m}}$ | |||||||||
3-16 | 34 | 0 | 1 |
blowup of 2-35 in the proper transform of a twisted cubic containing the center of the blowup |
2-27, 2-32, 2-35 | yes | yes | 0 | $\mathrm{B}$ | |||||||||
3-17 | 36 | 0 | 1 |
divisor on $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^2$ of degree $(1,1,1)$ |
* | 2-34 | yes | yes | 0 | $\mathrm{PGL}_2$ | ||||||||
3-18 | 36 | 0 | 1 |
blowup of 1-17 in the disjoint union of a line and a conic |
* | 2-29, 2-30, 2-33 | yes | yes | 0 | $\mathrm{B}\times\mathbb{G}_{\mathrm{m}}$ | ||||||||
3-19 | 38 | 0 | 1 |
blowup of 1-16 in two non-collinear points |
* | 2-35 | yes | yes | 0 | $\mathbb{G}_{\mathrm{m}}\times\mathrm{PGL}_2$ | ||||||||
3-20 | 38 | 0 | 1 |
blowup of 1-16 in the disjoint union of two lines |
2-31, 2-32 | yes | yes | 0 | $\mathbb{G}_{\mathrm{m}}\times\mathrm{PGL}_2$ | |||||||||
3-21 | 38 | 0 | 1 |
blowup of 2-34 in a curve of degree $(2,1)$ |
* | 2-34 | yes | yes | 0 | $\mathbb{G}_{\mathrm{a}}^2\rtimes\mathbb{G}_{\mathrm{m}}^2$ | ||||||||
3-22 | 40 | 0 | 1 |
blowup of 2-34 in a conic on $\{x\}\times\mathbb{P}^2$, $x\in\mathbb{P}^1$ |
2-34, 2-36 | yes | yes | 0 | $\mathrm{B}\times\mathrm{PGL}_2$ | |||||||||
3-23 | 42 | 0 | 1 |
blowup of 2-35 in the proper transform of a conic containing the center of the blowup |
2-30, 2-31, 2-35 | yes | yes | 0 | $\mathbb{G}_{\mathrm{a}}^3\rtimes(\mathrm{B}\times\mathbb{G}_{\mathrm{m}})$ | |||||||||
3-24 | 42 | 0 | 1 |
the fiber product of 2-32 with $\mathrm{Bl}_p\mathbb{P}^2$ over $\mathbb{P}^2$
|
* | 2-32, 2-34 | yes | yes | 0 | $\mathrm{PGL}_{3;1}$ | ||||||||
3-25 | 44 | 0 | 1 |
blowup of 1-17 in the disjoint union of two lines
|
* | 2-33 | yes | yes | 0 | $\mathrm{PGL}_{(2,2)}$ | ||||||||
3-26 | 46 | 0 | 1 |
blowup of 1-17 in the disjoint union of a point and a line
|
* | 2-34, 2-35 | yes | yes | 0 | $\mathbb{G}_{\mathrm{a}}^3\rtimes(\mathrm{GL}_2\times\mathbb{G}_{\mathrm{m}})$ | ||||||||
3-27 | 48 | 0 | 2 |
$\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ |
* | yes | yes | 0 | $\mathrm{PGL}_2^3$ | |||||||||
3-28 | 48 | 0 | 1 |
$\mathbb{P}^1\times\mathrm{Bl}_p\mathbb{P}^2$ |
* | 2-34 | yes | yes | 0 | $\mathrm{PGL}_2\times\mathrm{PGL}_{3;1}$ | ||||||||
3-29 | 50 | 0 | 1 |
blowup of 2-35 in a line on the exceptional divisor |
2-35 | yes | yes | 0 | $\mathrm{PGL}_{4;3,1}$ | |||||||||
3-30 | 50 | 0 | 1 |
blowup of 2-35 in the proper transform of a line containing the center of the blowup
|
* | 2-33, 2-35 | yes | yes | 0 | $\mathrm{PGL}_{4;2,1}$ | ||||||||
3-31 | 52 | 0 | 1 |
blowup of the cone over a smooth quadric in $\mathbb{P}^3$ in the vertex
|
* | yes | yes | 0 | $\mathrm{PSO}_{6;1}$ |
Fano threefolds with $\rho=4$
ID | $-\mathrm{K}_X^3$ | $\mathrm{h}^{1,2}$ | description | blowups | blowdowns | rational | unirational | moduli | $\mathrm{Aut}^0$ | ||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
4-1 | 24 | 1 |
divisor on $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ of degree $(1,1,1,1)$ |
3-27 | yes | yes | 3 | $0$ | |||||
4-2 | 28 | 1 |
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 |
3-31 | yes | yes | 2 | $\mathbb{G}_{\mathrm{m}}$ | |||||
4-3 | 30 | 0 |
blowup of 3-27 in a curve of degree $(1,1,2)$ |
3-17, 3-27, 3-28 | yes | yes | 0 | $\mathbb{G}_{\mathrm{m}}$ | |||||
4-4 | 32 | 0 |
blowup of 3-19 in the proper transform of a conic through the points |
* | 3-18, 3-19, 3-30 | yes | yes | 0 | $\mathbb{G}_{\mathrm{m}}^2$ | ||||
4-5 | 32 | 0 |
blowup of 2-34 in the disjoint union of a curve of degree $(2,1)$ and a curve of degree $(1,0)$ |
3-21, 3-28, 3-31 | yes | yes | 0 | $\mathbb{G}_{\mathrm{m}}^2$ | |||||
4-6 | 34 | 0 |
blowup of 1-17 in the disjoint union of 3 lines
|
3-25, 3-27 | yes | yes | 0 | $\mathrm{PGL}_2$ | |||||
4-7 | 36 | 0 |
blowup of 2-32 in the disjoint union of a curve of degree $(0,1)$ and a curve of degree $(1,0)$ |
3-24, 3-28 | yes | yes | 0 | $\mathrm{GL}_2$ | |||||
4-8 | 38 | 0 |
blowup of 3-27 in a curve of degree $(0,1,1)$ |
3-27, 3-31 | yes | yes | 0 | $\mathrm{B}\times\mathrm{PGL}_2$ | |||||
4-9 | 40 | 0 |
blowup of 3-25 in an exceptional curve of the blowup |
* | 3-25, 3-26, 3-28, 3-30 | yes | yes | 0 | $\mathrm{PGL}_{(2,2);1}$ | ||||
4-10 | 42 | 0 |
$\mathbb{P}^1\times\mathrm{Bl}_2\mathbb{P}^2$ |
* | 3-27, 3-28 | yes | yes | 0 | $\mathrm{PGL}_2\times\mathrm{B}^2$ | ||||
4-11 | 44 | 0 |
blowup of 3-28 in $\{x\}\times E$, $x\in\mathbb{P}^1$ and $E$ the $(-1)$-curve |
* | 3-28, 3-31 | yes | yes | 0 | $\mathrm{B}\times\mathrm{PGL}_{3;1}$ | ||||
4-12 | 46 | 0 |
blowup of 2-33 in the disjoint union of two exceptional lines of the blowup |
* | 3-30 | yes | yes | 0 | $\mathbb{G}_{\mathrm{a}}^4\rtimes(\mathrm{GL}_2\times\mathbb{G}_{\mathrm{m}})$ | ||||
4-13 | 26 | 0 |
blowup of 3-27 in a curve of degree $(1,1,3)$ |
3-27, 3-31 | yes | yes | 1 |
|
Fano threefolds with $\rho=5$
ID | $-\mathrm{K}_X^3$ | $\mathrm{h}^{1,2}$ | description | blowups | blowdowns | rational | unirational | moduli | $\mathrm{Aut}^0$ |
---|---|---|---|---|---|---|---|---|---|
5-1 | 28 | 0 |
blowup of 2-29 in the disjoint union of three exceptional lines of the blowup |
4-4, 4-12 | yes | yes | 0 | $\mathbb{G}_{\mathrm{m}}$ | |
5-2 | 36 | 0 |
blowup of 3-25 in the disjoint union of two exceptional lines on the same irreducible component |
4-9, 4-11, 4-12 | yes | yes | 0 | $\mathbb{G}_{\mathrm{m}}\times\mathrm{GL}_2$ | |
5-3 | 36 | 0 |
$\mathbb{P}^1\times\mathrm{Bl}_3\mathbb{P}^2$ |
* | 4-10 | yes | yes | 0 | $\mathrm{PGL}_2\times\mathbb{G}_{\mathrm{m}}^2$ |
Fano threefolds with $\rho=6$
ID | $-\mathrm{K}_X^3$ | $\mathrm{h}^{1,2}$ | description | blowups | blowdowns | rational | unirational | moduli | $\mathrm{Aut}^0$ |
---|---|---|---|---|---|---|---|---|---|
6-1 | 30 | 0 |
$\mathbb{P}^1\times\mathrm{Bl}_4\mathbb{P}^2$ |
* | 5-3 | yes | yes | 0 | $\mathrm{PGL}_2$ |
Fano threefolds with $\rho=7$
ID | $-\mathrm{K}_X^3$ | $\mathrm{h}^{1,2}$ | description | blowups | blowdowns | rational | unirational | moduli | $\mathrm{Aut}^0$ |
---|---|---|---|---|---|---|---|---|---|
7-1 | 24 | 0 |
$\mathbb{P}^1\times\mathrm{Bl}_5\mathbb{P}^2$ |
* | 6-1 | yes | yes | 2 | $\mathrm{PGL}_2$ |
Fano threefolds with $\rho=8$
ID | $-\mathrm{K}_X^3$ | $\mathrm{h}^{1,2}$ | description | blowups | blowdowns | rational | unirational | moduli | $\mathrm{Aut}^0$ |
---|---|---|---|---|---|---|---|---|---|
8-1 | 18 | 0 |
$\mathbb{P}^1\times\mathrm{Bl}_6\mathbb{P}^2$ |
* | 7-1 | yes | yes | 4 | $\mathrm{PGL}_2$ |
Fano threefolds with $\rho=9$
ID | $-\mathrm{K}_X^3$ | $\mathrm{h}^{1,2}$ | description | blowups | blowdowns | rational | unirational | moduli | $\mathrm{Aut}^0$ |
---|---|---|---|---|---|---|---|---|---|
9-1 | 12 | 0 |
$\mathbb{P}^1\times\mathrm{Bl}_7\mathbb{P}^2$ |
* | 8-1 | yes | yes | 6 | $\mathrm{PGL}_2$ |
Fano threefolds with $\rho=10$
ID | $-\mathrm{K}_X^3$ | $\mathrm{h}^{1,2}$ | description | blowdowns | rational | unirational | moduli | $\mathrm{Aut}^0$ |
---|---|---|---|---|---|---|---|---|
10-1 | 6 | 0 |
$\mathbb{P}^1\times\mathrm{Bl}_8\mathbb{P}^2$ |
9-1 | yes | yes | 8 | $\mathrm{PGL}_2$ |