## Rank 3 Fano 3-folds

- a double cover of branched along a divisor of tridegree (2,2,2). This is a hypersurface of type in the toric variety with weight data:

Quantum Lefschetz gives:

and regularizing gives period sequence 22:

Note that this is a G-Fano. - a member of on the -bundle over such that is irreducible, where is the tautological line bundle and is a member of . As discussed here, this is period sequence 97.
**We do not understand this variety**: in particular it does not seem to be Fano.Write for the ambient -bundle, which is the toric variety with weight data:

The line bundle is , so is a member of and . The equation defining takes the form:

where a linear function of ; and are homogeneous functions of of bidegrees (1,2) in the and ; and , , and are homogeneous functions of of bidegrees (2,3) in the and Consider now the subvariety defined by the equations in . This is a copy of ; note that on . The variety meets in the curve cut out by the equation inside . Without loss of generality we can take , so that on we have and . The curve is a copy of . We have that is trivial on (because the section of is non-vanishing on ) and that is also trivial on (because the section of is non-vanishing on ). Thus is trivial on , and so is not Fano.

- a divisor on of tridegree . This is straightforward quantum Lefschetz:

Regularizing this gives period sequence 31:

- the blow-up of , the 2-to-1 cover of with branch locus a divisor of type (2,2), with center a smooth fiber of . The variety is a hypersurface of type in the rank-2 toric variety with weight data:

We need to blow up the locus , obtaining our variety as a hypersurface of type in the toric variety with weight data:

Quantum Lefschetz gives:

and regularizing gives period sequence 151:

- the blow-up of with center a curve of bidegree (5,2) that projects isomorphically to a conic in . We construct this as a codimension-2 complete intersection in . has weight data:

The equations defining are

where are quadratic polynomials in the , and so is a complete intersection of type in . Quantum Lefschetz gives:

and regularizing gives a period sequence that we do not have yet:

- the blow-up of with center the disjoint union of a line and an elliptic curve of degree 4. Since is a (2,2) complete intersection in , our variety is a hypersurface of type in the toric variety with weight data:

Quantum Lefschetz gives:

and regularizing gives period sequence 146:

- the blow-up of with center an elliptic curve which is a complete intersection of two members of . Recall that is a (1,1) hypersurface in , and so is a complete intersection in of type . Quantum Lefschetz gives:

and regularizing gives period sequence 36:

Note that this is a D4 form (i.e. the Picard–Fuchs equation has unexpectedly low degree in D). It is almost certainly a G-Fano, as there is an obvious -action.

- a member of the linear system on , where are the projections and is the blow-up. The weight data for are:

Quantum Lefschetz gives:

and regularizing gives period sequence 85:

admits an obvious map to . Writing the equation defining in the form we find that is a divisor of type (0,2) and is a divisor of type (1,2). Thus is also the blow up of in a curve of bidegree (4,2) that is a complete intersection of type . - the blow-up of the cone over the Veronese surface with center the disjoint union of a vertex and a quartic in . Thus is a hypersurface of type in the toric variety with weight data:

Quantum Lefschetz gives:

and regularizing gives period sequence 68:

- the blow-up of a quadric in with center a disjoint union of two conics on . We take to be the locus in , and take the conics to be cut out of by and ; note that the intersection of these two planes misses . So we can construct the variety as a hypersurface of type in the toric variety with weight data:

Quantum Lefschetz gives:

and regularizing gives period sequence 67:

- the blow-up of with center a complete intersection of two general members of . is also the blow up of in a curve of bidegree (2,3) that is a complete intersection of type . Consider the toric variety with weight data:

This is , and is cut out here as a hypersurface of type . Quantum Lefschetz gives:

and regularizing gives period sequence 107:

To see that is a blow-up of as claimed, note that admits an obvious map to . Rewriting the equation defining in the form we see that is a divisor of type (1,1) and is a divisor of type (1,2). This suffices.

- the blow-up of with centre a disjoint union of a twisted cubic and a line. As we will see, is also the blow up of in a curve of bidegree (3,2) that projects isomorphically to a conic in . We begin by exhibiting as a complete intersection in a toric variety . The twisted cubic is cut out of by the equations:

The blow up of along is cut out of by the equation:

Observe that is disjoint from the line . We therefore blow up along the locus , obtaining the toric variety with weight data:

The equations defining inside are:

and so is a complete intersection of type . Quantum Lefschetz gives:

and regularizing gives period sequence 144:

To see that is a blow-up of as claimed, note that the map is . Rewriting the equations of as:

we see that is the blow-up of along the curve given by:

The equations defining are:

and so lies entirely within the “cylinder surface” . This cylinder surface is abstractly isomorphic to , where . The equations of become:

Thus is as described above. - the blow-up…
- the blow-up of with center the union of a cubic in a plane and a point not in . Let be in , and let be . Thus is the blow-up of the curve given by:

where the blow-up is . Thus is a hypersurface of type in the toric variety with weight data:

Quantum Lefschetz gives:

and regularizing gives period sequence 148:

- the blow-up of a quadric with center the disjoint union of a line on and a conic on . is the blow-up of along a curve of bidegree (2,2) that is a complete intersection of type . To see this, we first exhibit as a hypersurface in a toric variety. Note that the blow-up of along the plane and the line is toric with weight data:

The map to here sends ; there is also a map to given by . is cut out of the above toric variety by a section of . Thus we have:

and regularizing gives period sequence 112:

To see that is the blow-up of as claimed, write the equation defining as . Then by homogeneity is a section of (0,2) and is a section of (1,1). - the blow-up of with center the strict transform of a twisted cubic through the blown-up point . Let be the point in , and let be the curve given by

Let the blow-up be given by . Then the strict transform of in is given by:

As before, we introduce new variables and the toric variety with weight data:

is cut out by the equations:

Quantum Lefschetz gives:

and regularizing gives period sequence 119:

- a smooth divisor on of tridegree . This is straightforward quantum Lefschetz:

Regularizing this gives period sequence 37:

Note that is also the blow-up of along a curve of bidegree (1,2). - the blow-up of with center a disjoint union of a line and a conic. Take the line to be , and take the conic to be . The blow-up of in is , and the strict transform of the conic is cut out by . Thus is a hypersurface of type in the toric variety with weight data:

cut out by the equation . Quantum Lefschetz gives:

and regularizing gives period sequence 160:

- the blow-up of a quadric with center two non-colinear points. We construct this by taking the equation of the quadric to be inside , blowing up the line in (this line is not contained in ) and then taking the proper transform of the quadric inside the blow-up of . This is a hypersurface of type in the toric variety with weight data

Quantum Lefschetz gives:

and regularizing gives period sequence 57:

- the blow-up of a 3-dimensional quadric with center two disjoint lines on it. We take the quadric with equation in and blow up the lines and . Thus is a hypersurface of type in the toric variety with weight data:

Quantum Lefschetz gives:

and regularizing gives period sequence 63:

- the blow-up of along a curve of bidegree (2,1). is described by a single equation , where are quadratic polynomials in , in the toric variety with weight data

Quantum Lefschetz gives:

and regularizing gives period sequence 98:

- the blow-up of along a curve of bidegree (0,2), that is a conic in . Thus is described by a single equation in the toric variety with weight data

Quantum Lefschetz gives:

and regularizing gives period sequence 152:

- the blow-up of with center a conic passing through the blown-up point . Let be the point . Define the conic by:

Taking the proper transform under the blow-up gives the equations:

where the blow-up is , and so we need to consider the locus:

This is a hypersurface of type in the toric variety with weight data:

Quantum Lefschetz gives:

and regularizing gives period sequence 158:

- the fiber product where is a (1,1) hypersurface in . This is the blow up of along a curve of bidegree (1,1). To see this, note first that is cut out of by the equations:

The first equation here cuts out of ; the second equation cuts out of , as it is the equation defining the blow-up of .We now exhibit as the blow-up of a curve in by projecting to . This projection is an isomorphism away from the locus where the matrix

drops rank. This locus is:

i.e. a curve in of bidegree (1,1).

We can further simplify things by writing as a hypersurface in . Write the co-ordinates on as and the co-ordinates on as ; here the blow-up sends .The two equations defining (given above) reduce to the single equation:

Thus is a hypersurface of type in the toric variety with weight data:

Quantum Lefschetz gives:

and regularizing gives period sequence 86:

- the blow-up of with center two disjoint lines. This is the toric variety with weight data:

The blow-up map is . We have:

and regularizing gives period sequence 41:

- the blow-up of with center a disjoint union of a point and a line. This is also the blow-up of in a curve of bidegree (0,1). So is a toric variety, with weight data:

The blow-up map is . We have:

and regularizing gives period sequence 113:

- . This gives:

and regularizing gives period sequence 21:

- . This gives:

and regularizing gives period sequence 90:

- the blow-up of with center a line on the exceptional divisor of the blow-up . This is a toric variety with weight data:

We have:

and regularizing gives period sequence 163:

- the blow-up of with center the strict transform of a line through the blown-up point . This is a toric variety with weight data:

We have:

and regularizing gives period sequence 84:

- the total space of the bundle over . This is the toric variety with weight data:

We have:

and regularizing gives period sequence 53:

## Note

Consider the hypersurface of type in the toric variety with weight data:

We have:

and regularizing gives period sequence 32:

It seems that Mori-Mukai may have missed this variety, and have included number 2 in the rank 3 list by mistake. Note that our is a section of where is the tautological bundle on . The degree of is 28.

Never mind: this is #2 on the Mori-Mukai list of rank-4 Fanos.

On 25, 29, 30 and 31: these are smooth toric varieties.

They correspond to period sequences 41, 53, 84 and 163.

To be more precise:

25 is unique smooth toric Fano with P=3 and degree 44,

it corresponds to period sequence 41.

31 is unique smooth toric Fano with P=3 and degree 52,

it corresponds to period sequence 53.

29 and 30 has same degree 50 and correspond to p.s. 84 (grdb – 520136) and 163 (grdb – 520127).

Need an extra computation to separate these two.

Here is it.

Mirror of is

is a blowup of point, its mirror is

.

2.29 is blowup of line on exceptional divisor, so its mirror is

This Laurent polynomial has period sequence 163

[1, 0, 2, 0, 30, 60, 380, 840, 5950, 22680]

2.30 is blowup of line that strict transform of line passing the center of blowup ,

so its mirror is

This Laurent polynomial has period sequence 84

[1, 0, 2, 6, 30, 60, 470, 1680, 7630, 34440]

[Sorry, I had a typo here (!), 84 is correct]

Also, for 2.25 mirror is

On 3.7

It is not a G-Fano, but has a little of symmetry (), so it looks like

Fano with Picard number 2.

(According to Matsuki’s data) other Fanos with this property in this list should have the following numbers:

3, 9, 10, 17, 19, 20, 25, 31

I wrote a wider review of these issues in the post

expected distribution of equations

—-

Also I propose not to call this type of equation D4, since D4 is already reserved for equations that look like RQDE of P^4 (or 4-dimensional quadric). Better name for RQDE

of general Fano 3-fold with Picard number 2 is D3+1.

In general, type of RQDE for generic Fano variety is classified by its Lefschetz decomposition i.e. partition or Young tableux.

For Fano threefolds we will probably have just D3, D3+1, D3+2 and four D3+3’s.

(3+2 is a nickname for 3+2×1,

3+3 is a nickname for 3+3×1).

On 6, 10 and 23.

3.6 has period sequence 146

It is represented by complete intersection of degrees (1,0,2) and (0,1,1) in .

3.10 has period sequence 67.

It is represented by complete intersection of degrees (1,0,1), (0,1,1) and (0,0,2) in .

3.23 has period sequence 86.

It is represented by complete intersection of degrees (1,1,0) and (0,1,1) in .

On 3.2: its description basically says it is a divisor in smooth toric fourfold.

On 3.24:

there is a typo in the very end (!)

the period

is _not_ ps[130]: [1, 0, 4, 6, 60, 180, 1210, 5040, 30940, 150360]but ps[86]: [1, 0, 4, 6, 60, 180, 1210, 5460, 30940, 165480]

They have same first 7 entries, so it was easy to mistake.

Moreover, period sequence 130 is bad – it has (2D-1) as a multiple. So this resolves the ambiguity we had before.

Also – 3.24 has 3 terminal Gorenstein degenerations,

and TGTD ansatz applied to them is exactly ps[86].

3.24 corresponds to ps[86]

3.23 should correspond to ps[158]

And there will be exactly 1-1 correspondence between 98 Fano threefolds and 98 good period sequences.

Following is the computation of first 10 terms:

Moreover, the description for 3.24 seems too complicated.

After on the third line X is described as the complete intersection of degrees (1,1,0) and (0,1,1) in we can already apply quantum Lefschetz:

,

regularizing and normalizing we get

i.e. period sequence 86.