How it works

How to use it

In Step 1, choose an OS table number between 1 and 74. These are the entries in Oguiso and Shioda’s classification of the possible Mordell–Weil lattices of rational elliptic surfaces (Oguiso–Shioda, Main Theorem). The explorer will display the singular fibers in their chosen order, together with the Mordell–Weil group and the number of generators used by the program. The fiber collision profile selector covers all 279 configurations in Persson–Miranda's classification, represented by 289 OS/profile choices. Sometimes the same configuration occurs multiple times, with different Mordell–Weil groups.

Once the surface S has loaded, move to Step 2. Choose a section P, expressed as an integral linear combination of the displayed generators. One can think of this as constructing the principal ℂ^*-bundle M^×, where M = O_S(P−O). Next choose a second section Q. Translation by Q defines an automorphism t_Q of S.

At least one of P or Q must be narrow at every singular fiber: that is, at least one of the two sections must meet the identity component. The relevant congruence conditions are displayed on the right, together with the height-pairing matrix on MW(S).

When these conditions hold, Hom(t_Q^*M, M) ≃ O_ℙ¹(n), where n = ⟨P, Q⟩ is the Mordell–Weil height pairing; see, for example, Schütt–Shioda, §11. Click “Check sections” to verify the conditions and continue to Step 3.

We call Hom(t_Q^*M, M) the linearization bundle. A linearization is determined by a meromorphic section on S pulled back from ℙ¹. Depending on the sign of its degree, you choose either zeros or poles whose orders sum to the required total degree. In the present implementation, zeros and poles may be placed only over smooth fibers or fibers of Kodaira type I_k. Additional smooth I₀ fibers may be added for this purpose or for the following step.

Away from its zeros or poles, the linearization lifts t_Q from S to an automorphism of M^×. One takes the quotient of the appropriate open subset of M^× by the ℤ-action generated by this lift, then fills in the family at each zero or pole using a Mumford construction. The prescribed models are either twisted wheels over an elliptic curve in the rank one case or honeycombs of degree six del Pezzo surfaces arising from the periodic A₂-tiling. See Engel–Filipazzi–Greer–Mauri–Svaldi, Definition 6.8 for twisted n-wheels and Engel–de Gaay Fortman–Schreieder, Construction 3.3 for the Mumford construction.

In Step 4, choose the logarithmic modifications. These are represented by locally monodromy-invariant vectors in the rational lattice H₁(T, ℚ) ≃ ℚ⁴ of a nearby complex two-torus. The denominator of such a vector records the order of the log transform. In the current implementation, this order must divide the degree of the base change required for semistable reduction, unless the fiber is smooth, in which case, a log transform of arbitrary torsion order is permitted.

The log-transform datum is entered as a sequence of rational numbers. You may use either:

the displayed basis of the integral monodromy-invariant lattice; or
the original ambient basis of the rank four lattice.

The first format is strongly recommended, since an arbitrary vector in the ambient basis will usually fail to be monodromy invariant.

For a fractional log transform at a smooth fiber, use an I₀ slot with linearization weight zero. A slot supporting a zero or pole has a Mumford filling; add a separate smooth slot for the fractional twist.

A log transform twists the semistable base change action by a torsion translation. The resulting quotient is filled using one of the precomputed smooth birational models in the local-model database. At a potentially good additive fiber or an I_n* fiber, selecting “None” retains the original filling, while an explicit zero vector selects the quotient of the semistable reduction model with zero additional twist; these need not be the same. At an I_k fiber, including I₀, the two choices agree.

Once all the data have been entered, press “Compute topology.” The example selector at the top of the page can populate the form with several constructions of S⁶, illustrating the expected input formats. The first pre-loaded example corresponds to Alpöge–Claude and Engel, and the second and third to work of Viaclovsky and Liu–Rollenske, respectively. There is a celebration if you produce S⁶!

The program first computes the monodromy of the underlying elliptic surface. Its initial representatives come from Fukae, Table 3 and §5, organized according to the Oguiso–Shioda classification. The additional collision profiles are determined via genus zero branched covers of the j-line and quadratic twists, following Miranda, §3, with direct models for constant j. For the newly added profiles, the monodromy matrices and data about sections are precomputed. See the collision catalogue notes.

It then lifts the elliptic monodromy to the rank four integral local system of the complex two-torus fibration. The lift is computed through the natural filtration associated with the principal ℂ^*-bundle, using the local specialization data of P and Q, together with the order of the zero or pole of the linearization. The resulting formula is closely related to equation (4.9) of Viaclovsky.

The topology of 865 tabulated local models is stored in a read-only database. For each model, the database records an integral cochain complex for a tubular neighborhood of the singular fiber, a compatible model of its boundary, and the comparison with the marked cohomology of a nearby smooth fiber.

The local cochain models are combined with a cochain model for the smooth part of the torus fibration. This produces the Mayer–Vietoris complex associated with the cover by the smooth locus of the fibration and the tubular neighborhoods of the singular fibers. Computing its cohomology gives the integral cohomology groups of the total space, including torsion and extension information.

The database also records the peripheral relations contributed by each filling. These relations are combined with the monodromy of the smooth fibration to perform the van Kampen computation of the fundamental group—in other words, to determine which loops are killed when each singular fiber is filled in.

Detailed derivations of the local models and of the marked cochain comparisons are maintained separately and are not currently published with the applet.

Possible extensions

The following cases are not included in the current implementation:

log transforms at singular fibers, whose order does not divide the degree of the base change required for semistable reduction;
zeros or poles of the linearization bundle supported on additive Kodaira fibers;
Mumford subdivisions other than the prescribed A₂ model, or models outside the bounds of the current database;
general recognition of arbitrary finitely presented fundamental groups.