I was reading an amusing paper of Ronald van Luijk: he demonstrates
the K3 surface with Picard number one.
Reflecting his proof I formulate the following unexpected principle:
in order to classify Fanos with Picard number 1
we should NOT just get rid of all polytopes with higher Picard number !
One of the reasons is as follows:
0. Assume we’ve done with computing basic invariants of polytopes,
including Picard numbers
1. Assume we’ve done with the initial step of creating some database
of prospective Laurent polynomials out of many many polytopes
2. It is much easier to create a “phone book” of period sequences (say
first 10 or 20 coefficients) then to compute their Picard–Fuchs
operators.
3. Once we have a “phone book” prospective polynomials W fall into
equivalence classes with respect to initial terms of their period
sequences
If we manage how to program Cremona-equivalence then they fall
into even better classes.
4. Now, under assumption that Laurent polynomial W reflects a toric
degeneration of Fano manifold X into the toric variety X_0 that W is
supported on (i.e. Newton(W) is dual to Moment(X_0))
$ rk Pic X \geq \rk \Pic X_0 $
5. So if we know that period sequence is supported on some toric
variety with Picard number > 1 – then it is not a period sequence of
Fano with Picard number 1
If we would drop polytopes with Picard number >1, then we have to
do extra work!
In fact we used this ideas for threefolds before we computed their
G-series by ad hoc and a posteriori methods.
By a posteriori I mean it is not so easy to tell what is Picard number
of Fano variety from its G-series and/or differential operator that
annihilates it:
G-series for Fano varieties with Picard number 1 and for G-minimal
(hence quantum minimal) varieties look pretty much the same.