Moduli of weighted stable elliptic surfaces and invariance of log plurigenera

Abstract

Motivated by Hassett's weighted pointed stable curves, we use the log minimal model program to construct compact moduli spaces parameterizing weighted stable elliptic surfaces — elliptic fibrations with section and marked fibers each weighted between zero and one. Moreover, we show that the domain of weights admits a wall and chamber structure, describe the induced wall‐crossing morphisms on the moduli spaces as the weight vector varies, and describe the surfaces that appear on the boundary of the moduli space. The main technical result is a proof of invariance of log plurigenera for slc elliptic surface pairs with arbitrary weights.

Wall crossing for moduli of stable log pairs

“…which one can think of as a sort of universal generalized log flip (see [AB20,Proposition 8.4] and the preceding discussion). Indeed this diagram pulling back along the natural morphism B → F yields the generalized log flip…”

“…This provides alternate compactifications of the spaces of Hacking-Keel-Tevelev [HKT06]. Similarly, in [AB20] compact moduli spaces of weighted stable elliptic surfaces are constructed (see also [Inc20]). These moduli spaces parametrize pairs of an elliptic surface with the divisor consisting of a section and some weighted (possibly singular) fibers.…”

“…Explicit stable pair compactifications of moduli of higher dimensional varieties have been studied extensively in recent years, e.g. weighted hyperplane arrangements [HKT06,Ale15], principally polarized abelian varieties [Ale02], plane curves [Hac04], and elliptic surfaces [AB20,Inc20], etc.…”

Smoothability of relative stable maps to stacky curves

“…In [AB19, Section 1.4] we proposed the problem of using log geometry to give the main component a moduli theoretic interpretation and classify the boundary components. The present paper solves this problem for elliptic fibrations with marked singular fibers (we refer the reader to [AB21] and [AB20, Section 4]). A key observation is Conditions ( * ) relative to ∞ ∈ M 1,1 as well as the choice of stabilizers on the marked points translate to conditions on the configuration of singular fibers on the components of the twisted elliptic surface ([AB20, Propositions 4.1 & 4.4]).…”

Explicit Constructions of Halphen Pencils

“…Rational elliptic surfaces admitting a global section have been widely studied under many different points of view. Different compactifications for their moduli space have been constructed and are well understood [1], [3], [4], [22], [33], [34]; their automorphism groups have been classified [24], [25]; and all possible configurations of singular fibers are known [36], [40]. It is also known that these surfaces can be realized from a pencil of cubic curves in the plane (by blowing-up their nine base points) and explicit examples having a Mordell-Weil group with some particular rank have been considered in [13,Theorem 5.6.2], [19], [39] and [42].…”