Expected distribution of equations.

Let X be a smooth Fano threefold with Picard number P = \rho = dim H^2(X).

Then subring of algebraic (even) cycles in X is (2+2P)-dimensional, and its Lefschetz decomposition has P blocks: 1 block of length 4 and (P-1) blocks of length 2. So its image in cohomologies of anticanonical section (K3 surface) is (2+2P – P) = (2+P)-dimensional.

For “general” Fano threefold with Picard number P we expect
regularized quantum differential equation (RQDE) to be of degree (2+P) in D = t \frac{d}{dt}
and to have (2+2P) singular points. Nevertheless degree in t may be more than number of singular points
due to apparent singularities.

It turns out that condition for general is not very general in practice.

Assume Fano threefold X has action of finite group G in one of the 4 ways:
a. G acts on X by regular (algebraic) transformations,
b. G acts on X by symplectic transformations,
c. X is defined over non-algebraically closed field k and G is Galois group Gal(k),
d. X is a fiber of a smooth family over some base B and fundamental group G = \pi_1(B) acts on H^\bullet(X) via monodromy.

For cases a,b,c consider the induced action of G on cohomology of X.

Let p = \rho^G = dim H^2(X)^G be $G$-invariant Picard number.
G-invariant part of cohomologies H(X)^G is (2+2p)-dimensional.

Define \emph{minimal quantum cohomology subring } QH_m(X) of \emph{very small quantum cohomology ring} QH(X) as subring generated by c_1(X) and C[t].
It is easy to see QH_m(X) is contained in H(X, C)^G [t].

This implies that regularized I-series I_X is annihilated by
differential operator of degree (2+p).

So it is natural to ask about possible G-actions on Fano threefolds.
First (numerical) step is to see the possible automorphisms of Mori cone or Kaehler cone.
We have some structures on H^2(X,R):
a. lattice H^2(X,Z) and element c_1(X) inside the lattice,
b. rational polyhedral cone of numerically effective divisors,
c. nondegenerate integral quadratic form (Lefschetz pairing) : (A,B) -> \int_X A \cup B \cup c_1(X).
We call this information \emph{Mori structure}.

Group of automorphisms of Mori structure is finite, and for any action
G-invariant Picard number is not less than dimension of invariants of H^2(X)
with respect to whole group of automorphisms of Mori structure.

As far as I remember (but cannot find a reference) for all Fano threefolds one may find some moduli
and some kind of G-action such that G-invariant Picard group coincides with invariant part of $H^2$ with respect to automorphisms of Mori structure.

The standard reference for automorphisms of Mori structure is probably:
Kenji Matsuki, “Weyl groups and birational transformations among minimal models”, AMS 1995

He studies slightly different problem, but has a similar answer. Unfortunately I haven’t a copy of this book, but copied one page from google books.

He says automorphisms of Mori structures turn out to be Weyl groups.

He claims the following Fano threefolds have nontrivial automorphisms:

P – Picard number, then list of Mori-Mukai numbers with the given Picard number

A_1: 2, 6, 12, 21, 32 (these should be G-Fano, but number 2 is suspicious)
other have p=2

A_2: 1, 27 (G-Fano, suspicious that 13 is in the next line)
A_1: 3, 7, 9, 10, 13, 17, 19, 20, 25, 31 (should correspond to p=2)
other have p=3

A_3: 1 (G-Fano)
A_2: 6 (p=2)
A_1 \times A_1: 2 (p=2)
A_1: 3, 4, 7, 8, 10, 12 (p=3)
trivial – 5,9,11 (should have p=4)
missing number 13 from Erratum

A_1 \times A_2: 3 (p=2)
A_2: 1 (p=3)
A_1: 2 (p=4)

For cases P \geq 6 our threefolds are products of a line and del Pezzo surface P^1 \times S_d. They all have Weyl group of type E_{9-d} and p=2.

So the distribution in p is the following (case 4.13 is missing):
p is always less than 5;
p=4 – 4 varieties: 5.2; 4.5, 4.9, 4.11
p=3 – 26 varieties: 5.1; 4.3, 4.4, 4.7, 4.8, 4.10, 4.12; and 19 with P=3
p=2 – 50 varieties
p=1 – 25 varieties (or 26 if 2.2 is there)

This means just 4 varieties should have N=6, and other have even less.

The obvious thing to do is to recompute ourselves the Mori structure and its automorphisms
(in particular discriminant of Lefschetz quadratic form is an important invariant that we need anyway).

One Comment

  1. Sergey says:

    Since now we computed the minimal degrees of quantum differential equations for Fano threefolds ad hoc we can list all the discrepancies between our result and Matsuki’s. There are just 3 of them:

    2.2 – Matsuki has one more symmetry than us.
    3.13 – we have one more symmetry than Matsuki.
    4.7 – we have one more symmetry than Matsuki.

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