Slicing the data
Sergei: I sliced the Minkowski period data as you requested, here.
Update: here is a new version which also contains the Picard ranks.
A collaborative research blog.
Sergei: I sliced the Minkowski period data as you requested, here.
Update: here is a new version which also contains the Picard ranks.
Sergei: for
I get 
, 
. See period sequence 44.
OK, not tomorrow. Maybe the next day…
Tom: thank you very much, I figured out my disambiguation.
The problem is the apparent singularities.
Choose a coordinate
.
— a root of 
.
Choose
For
there are two singularities: 
and 
.
there are three singularities: 
, 
and 
.
For
In case of
we should have 6 singularities, in coordinate W these are the sums singularities for 
and for 
:
and 
, two roots of a^2 – a + 7 are 
and 
.
1=3-2, 5 = 3+2, two roots of a^2+7 a+19 are
In inverse to W coordinate t this should correspond to a product
in front of D^4.
However the symbol of PF equation is not exactly this degree 6 polynomial, but
, solutions of this degree 3 equation are apparent singularities – the monodromies around these points are identities.
degree 9 polynomial in t:
107730*t^9 – 211376*t^8 – 50139*t^7 + 156996*t^6 – 20520*t^5 + 16194*t^4 + 1782*t^3 – 448*t^2 – 243*t + 24
There is an extra factor of
So far I cannot think on an easy cure for these (not involving computing the monodromies).
However it could be useful to check for the decomposition (over integers) of the PF’s symbol – if the symbol is indecomposable and of high degree (more than 10 in general) then it is certainly suspicious.
For smooth Fano threefold with (G-invariant) Picard number
we expect N to be 
and r to be 
.
.
So we expect the symbol to have some factors of degree
For Picard ranks, could you please add both maximal and minimal ranks that appear?
When Fano has a degeneration to (Gorenstein/any) toric variety with Pic=Z ?
I’ve added a new version containing the Picard ranks.
Sergei: I grabbed these directly from grdb because I couldn’t get your gp code to work. Did you mean concat rather than matconcat?
Sorry about the delay — I was on holiday.
Tom: sorry, indeed I have alias matconcat=concat for backward compatibility.
Here are my .gpalias and .gprc files.
Generally degree
and Picard rank 
is not enough to determine the smooth Fano Y.
and
— discriminant of quadratic form 
on the lattice Pic(Y).
We need two more invariants:
For STD
and number b is given by the formula
where v and f are numbers of vertices and faces of fan polytope.
So number b should also be easily derived from grdb data.
Also for STD
.
However I don’t think there is any way to compute this number essentially simpler than the obvious one used in my script.
In case of degeneration to non-terminal Gorenstein variety X Picard number
may fall down.
If for some reason Picard number remains the same, then Pic(X) is a sublattice of Pic(Y) of some finite index r. Then
,
but since d(Y) is rarely divisible by a square this happens not so often.
Tom, could you please post a new version which contains both Picard ranks and degrees?
Freshly fixed, updated and improved script for computing the principal invariants of the smoothing