Dubrovin's conjecture
explain this
Picard rank $\rho$  1  2  3  4  5  6  7  8  9  10  total 

number of families  17  36  31  13  3  1  1  1  1  1  105 
number of families with $\mathrm{h}^{1,2}=0$  4  14  22  11  3  1  1  1  1  1  59 
number of families with generically semisimple small quantum cohomology  4  13  14  4  2  1  1  1  1  1  42 
number of families with full exceptional collections constructed  4  14  22  11  3  1  1  1  1  1  59 
For now, only generic semisimplicity of small quantum cohomology is in the table. The generic semisimplicity of quantum cohomology for all the known cases is established at the level of small quantum cohomology, which implies it at the big level (but not conversely). Once Fano 3folds where only the big quantum cohomology is known to be generically semisimple, the table will be updated accordingly.
ID  description  existence of a full exceptional collection  generic semisimplicity of small quantum cohomology 

110 
zero locus of $(\bigwedge^2\mathcal{U}^\vee)^{\oplus 3}$ on $\mathrm{Gr}(3,7)$ 
Kuznetsov constructed it in 1996 
someone proved it in at some point using 
115 
section of Plücker embedding of $\mathrm{Gr}(2,5)$ by codimension 3 subspace 
Orlov constructed it in 1991 
someone proved it in at some point using 
116 
hypersurface of degree 2 in $\mathbb{P}^4$ 
Kapranov constructed it in 1986 
someone proved it in at some point using 
117 
projective space $\mathbb{P}^3$ 
Beilinson constructed it in 1978 
someone proved it in at some point using Iritani proved it in 2007 using toric geometry 
220 
blowup of 115 in a twisted cubic 
constructed using Orlov's blowup formula  not known to hold 
221 
blowup of 116 in a twisted quartic 
constructed using Orlov's blowup formula 
Ciolli proved it in 2005 using the description of a 1 or 2curve blowup of $\mathbb{P}^3$ or $Q^3$ 
222 
blowup of 115 in a conic 
constructed using Orlov's blowup formula 
Ciolli proved it in 2005 using the description of a 1 or 2curve blowup of $\mathbb{P}^3$ or $Q^3$ 
224 
divisor on $\mathbb{P}^2\times\mathbb{P}^2$ of bidegree $(1,2)$ 
constructed using Orlov's projective bundle formula 
Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$bundle 
226 
blowup of 115 in a line 
constructed using Orlov's blowup formula 
Ciolli proved it in 2005 using the description of a 1 or 2curve blowup of $\mathbb{P}^3$ or $Q^3$ 
227 
blowup of 117 in a twisted cubic 
constructed using Orlov's blowup formula 
Ciolli proved it in 2005 using the description of a 1 or 2curve blowup of $\mathbb{P}^3$ or $Q^3$ Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$bundle 
229 
blowup of 116 in a conic 
constructed using Orlov's blowup formula 
Ciolli proved it in 2005 using the description of a 1 or 2curve blowup of $\mathbb{P}^3$ or $Q^3$ 
230 
blowup of 117 in a conic 
constructed using Orlov's blowup formula 
Ciolli proved it in 2005 using the description of a 1 or 2curve blowup of $\mathbb{P}^3$ or $Q^3$ 
231 
blowup of 116 in a line 
constructed using Orlov's blowup formula 
Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$bundle 
232 
divisor on $\mathbb{P}^2\times\mathbb{P}^2$ of bidegree $(1,1)$

constructed using Orlov's projective bundle formula 
Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$bundle 
233 
blowup of 117 in a line 
constructed using Orlov's blowup formula 
Ciolli proved it in 2005 using the description of a 1 or 2curve blowup of $\mathbb{P}^3$ or $Q^3$ Iritani proved it in 2007 using toric geometry 
234 
$\mathbb{P}^1\times\mathbb{P}^2$ 
constructed using Orlov's projective bundle formula 
Iritani proved it in 2007 using toric geometry Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$bundle 
235 
$\mathrm{Bl}_p\mathbb{P}^3$

constructed using Orlov's projective bundle formula 
Iritani proved it in 2007 using toric geometry Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$bundle 
236 
$\mathbb{P}(\mathcal{O}_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}(2))$ 
constructed using Orlov's projective bundle formula 
Iritani proved it in 2007 using toric geometry Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$bundle 
35 
blowup of 234 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 
constructed using Orlov's blowup formula  not known to hold 
38 
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 $\pi_i\colon\mathrm{Bl}_1\times\mathbb{P}^2$ are the projections, and $\alpha\colon\mathrm{Bl}_1\mathbb{P}^2\to\mathbb{P}^2$ is the blowup 
constructed using Orlov's blowup formula  not known to hold 
310 
blowup of 116 in the disjoint union of 2 conics

constructed using Orlov's blowup formula 
Ciolli proved it in 2005 using the description of a 1 or 2curve blowup of $\mathbb{P}^3$ or $Q^3$ 
312 
blowup of 117 in the disjoint union of a line and a twisted cubic 
constructed using Orlov's blowup formula 
Ciolli proved it in 2005 using the description of a 1 or 2curve blowup of $\mathbb{P}^3$ or $Q^3$ 
313 
blowup of 232 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 $i=1,2$ 
constructed using Orlov's blowup formula  not known to hold 
315 
blowup of 116 in the disjoint union of a line and a conic 
constructed using Orlov's blowup formula 
Ciolli proved it in 2005 using the description of a 1 or 2curve blowup of $\mathbb{P}^3$ or $Q^3$ 
316 
blowup of 235 in the proper transform of a twisted cubic containing the center of the blowup 
constructed using Orlov's blowup formula  not known to hold 
317 
divisor on $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^2$ of degree $(1,1,1)$ 
constructed using Orlov's blowup formula 
Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$bundle 
318 
blowup of 117 in the disjoint union of a line and a conic 
constructed using Orlov's blowup formula 
Ciolli proved it in 2005 using the description of a 1 or 2curve blowup of $\mathbb{P}^3$ or $Q^3$ 
319 
blowup of 116 in two noncollinear points 
constructed using Orlov's blowup formula  not known to hold 
320 
blowup of 116 in the disjoint union of two lines 
constructed using Orlov's blowup formula 
Ciolli proved it in 2005 using the description of a 1 or 2curve blowup of $\mathbb{P}^3$ or $Q^3$ 
321 
blowup of 234 in a curve of degree $(2,1)$ 
constructed using Orlov's blowup formula  not known to hold 
322 
blowup of 234 in a conic on $\{x\}\times\mathbb{P}^2$, $x\in\mathbb{P}^1$ 
constructed using Orlov's blowup formula  not known to hold 
323 
blowup of 235 in the proper transform of a conic containing the center of the blowup

constructed using Orlov's blowup formula  not known to hold 
324 
the fiber product of 232 with $\mathrm{Bl}_p\mathbb{P}^2$ over $\mathbb{P}^2$ 
constructed using Orlov's blowup formula 
Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$bundle 
325 
blowup of 117 in the disjoint union of two lines

constructed using Orlov's blowup formula 
Ciolli proved it in 2005 using the description of a 1 or 2curve blowup of $\mathbb{P}^3$ or $Q^3$ Iritani proved it in 2007 using toric geometry Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$bundle 
326 
blowup of 117 in the disjoint union of a point and a line

constructed using Orlov's blowup formula 
Iritani proved it in 2007 using toric geometry 
327 
$\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ 
constructed using Orlov's projective bundle formula 
Iritani proved it in 2007 using toric geometry Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$bundle 
328 
$\mathbb{P}^1\times\mathrm{Bl}_p\mathbb{P}^2$ 
constructed using Orlov's blowup formula 
Iritani proved it in 2007 using toric geometry Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$bundle 
329 
blowup of 235 in a line on the exceptional divisor 
constructed using Orlov's blowup formula 
Iritani proved it in 2007 using toric geometry 
330 
blowup of 235 in the proper transform of a line containing the center of the blowup

constructed using Orlov's blowup formula 
Iritani proved it in 2007 using toric geometry Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$bundle 
331 
blowup of the cone over a smooth quadric in $\mathbb{P}^3$ in the vertex

constructed using Orlov's projective bundle formula 
Iritani proved it in 2007 using toric geometry Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$bundle 
43 
blowup of 327 in a curve of degree $(1,1,2)$ 
constructed using Orlov's blowup formula  not known to hold 
44 
blowup of 319 in the proper transform of a conic through the points 
constructed using Orlov's blowup formula  not known to hold 
45 
blowup of 234 in the disjoint union of a curve of degree $(2,1)$ and a curve of degree $(1,0)$ 
constructed using Orlov's blowup formula  not known to hold 
46 
blowup of 117 in the disjoint union of 3 lines

constructed using Orlov's blowup formula  not known to hold 
47 
blowup of 232 in the disjoint union of a curve of degree $(0,1)$ and a curve of degree $(1,0)$ 
constructed using Orlov's blowup formula  not known to hold 
48 
blowup of 327 in a curve of degree $(0,1,1)$ 
constructed using Orlov's blowup formula  not known to hold 
49 
blowup of 325 in an exceptional curve of the blowup 
constructed using Orlov's blowup formula 
Iritani proved it in 2007 using toric geometry 
410 
$\mathbb{P}^1\times\mathrm{Bl}_2\mathbb{P}^2$ 
constructed using Orlov's blowup formula 
Iritani proved it in 2007 using toric geometry Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$bundle 
411 
blowup of 328 in $\{x\}\times E$, $x\in\mathbb{P}^1$ and $E$ the $(1)$curve 
constructed using Orlov's blowup formula 
Iritani proved it in 2007 using toric geometry 
412 
blowup of 233 in the disjoint union of two exceptional lines of the blowup 
constructed using Orlov's blowup formula 
Iritani proved it in 2007 using toric geometry 
413 
blowup of 327 in a curve of degree $(1,1,3)$ 
constructed using Orlov's blowup formula  not known to hold 
51 
blowup of 229 in the disjoint union of three exceptional lines of the blowup 
constructed using Orlov's blowup formula  not known to hold 
52 
blowup of 325 in the disjoint union of two exceptional lines on the same irreducible component 
constructed using Orlov's blowup formula 
Iritani proved it in 2007 using toric geometry 
53 
$\mathbb{P}^1\times\mathrm{Bl}_3\mathbb{P}^2$ 
constructed using Orlov's blowup formula 
Iritani proved it in 2007 using toric geometry Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$bundle 
61 
$\mathbb{P}^1\times\mathrm{Bl}_4\mathbb{P}^2$ 
constructed using Orlov's blowup formula 
Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$bundle 
71 
$\mathbb{P}^1\times\mathrm{Bl}_5\mathbb{P}^2$ 
constructed using Orlov's blowup formula 
Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$bundle 
81 
$\mathbb{P}^1\times\mathrm{Bl}_6\mathbb{P}^2$ 
constructed using Orlov's blowup formula 
Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$bundle 
91 
$\mathbb{P}^1\times\mathrm{Bl}_7\mathbb{P}^2$ 
constructed using Orlov's blowup formula 
Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$bundle 
101 
$\mathbb{P}^1\times\mathrm{Bl}_8\mathbb{P}^2$ 
constructed using Orlov's blowup formula 
Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$bundle 