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 bundleis
, so
is a member of
and
. The equation defining
takes the form:
wherea 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 definingare
whereare 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 ofalong
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 defininginside
are:
and sois 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 thatis the blow-up of
along the curve
given by:
The equations definingare:
and solies entirely within the “cylinder surface”
. This cylinder surface is abstractly isomorphic to
, where
. The equations of
become:
Thusis 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-upis
. 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 tohere 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 thatis 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-upbe given by
. Then the strict transform of
in
is given by:
As before, we introduce new variablesand 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 thatis 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-upgives the equations:
where the blow-up is, and so we need to consider the locus:
This is a hypersurface of typein 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 cutsout 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 inof bidegree (1,1).
We can further simplify things by writingas 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:
Thusis 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 ![w_{\PP^3} = x+y+z+\frac{1}{xyz} w_{\PP^3} = x+y+z+\frac{1}{xyz}](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/569/569abb3f527abdcc1393dd618ee4f303-ffffff-000000-0.png)
2.29 is blowup of line on exceptional divisor, so its mirror is
![w_{2.29} = w_{V_7} + x^2 yz = ((x+y+z+\frac{1}{xyz}) + xyz) +x^2 yz w_{2.29} = w_{V_7} + x^2 yz = ((x+y+z+\frac{1}{xyz}) + xyz) +x^2 yz](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/a6e/a6e3538f251fe5633824a483f04ba6a7-ffffff-000000-0.png)
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
,
![w_{2.30} = w_{V_7} + xy = ((x+y+z+\frac{1}{xyz}) + xyz) + xy w_{2.30} = w_{V_7} + xy = ((x+y+z+\frac{1}{xyz}) + xyz) + xy](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/47d/47d0b7ffb7dc722f68d1e5287f7eac26-ffffff-000000-0.png)
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![w_{2.25} = (x+y+z+\frac{1}{xyz}) + xy + \frac{1}{xy} w_{2.25} = (x+y+z+\frac{1}{xyz}) + xy + \frac{1}{xy}](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/bb4/bb487e7728b7794a086aa82f46a962e7-ffffff-000000-0.png)
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.
we can already apply quantum Lefschetz:
After on the third line X is described as the complete intersection of degrees (1,1,0) and (0,1,1) in
regularizing and normalizing we get
i.e. period sequence 86.