Modular compactifications of ℳ2,n with Gorenstein curves

A smooth compactification of the space of genus two curves in projective space: via logarithmic geometry and Gorenstein curves

“…The resulting space of admissible covers is as nice as the moduli space of curves, but it has the advantage of encoding the Brill-Noether theory of the curve in the logarithmic structure; see S Mochizuki [55]. The necessity of such enrichment can be understood already by looking at the isolated Gorenstein singularities of genus two (see Battistella [11]): there are two families of these, basically corresponding to the choice of either a Weierstrass point or a conjugate pair in the semistable model. In order to tell these two cases apart logarithmically, in the construction of C !…”

“…In [11], the first author produces a sequence of alternative compactifications of M 2;n based on replacing genus one and two subcurves with few special points by isolated Gorenstein singularities. Although we do not discuss it here, the techniques we develop also provide a resolution of the rational maps among these spaces.…”

Curve counting in genus one: Elliptic singularities and relative geometry

“…• There are several compactifications of M g,n parametrising only Gorenstein curves: the most notable one is the Deligne-Mumford compactification M g,n , and the alternatives serve to illustrate its birational geometry, see for instance [HH09,Smy11a,AFSvdW17,Bat22,BKN23]. Indeed, all singularities appearing in the Hassett-Keel program seem to be Gorenstein [AFS16].…”

“…Isolated Gorenstein singularities of genus one have been classified by Smyth in [Smy11a] (with applications to moduli theory in [Smy11b,Smy19] and the other references above). Isolated Gorenstein singularities of genus two have been classified by the author in [Bat22]; some non-isolated singularities, that we call "ribbons with tails", appear in [BC23]. Smoothable hyperelliptic singularities have been classified in [BB23].…”

“…If Z is reducible, these multiplicities determine further the shape of the dual graph: for instance, when g = 1, all the p i must be at the same distance from the minimal genus one subcurve[Smy11a]. In general, these conditions can be phrased effectively in terms of logarithmic geometry, by requiring the existence of certain piecewise-affine functions on the dual graph, which turn out to be differentials in the sense of tropical geometry: see[Bat22] and §1.5 below. It was pointed out to me by F. Viviani that this suggests the following statement: the variety of stable tails[Has00] of a Gorenstein singularity is the Deligne-Mumford compactification of a stratum of differentials.…”