research

We compute the integral Chow ring of the moduli space of pointed elliptic curves with few (3 and 4) markings, leveraging the geometry of Smyth's alternative compactifications. After work of Lekili-Polishchuk, Smyth's space of (n-1)-stable n-pointed elliptic curves can be described via explicit equations, and it is simply a weighted projective stack for low values of n. This allows us to present the Deligne-Mumford compactification as a sequence of weighted blow-ups and blow-downs of a weighted projective space, thus reducing the computation of its Chow ring to a simple application of the weighted blow-up formula.

We construct a virtual class on the space of punctured maps which is compatible with splitting the source curve of a log stable map. We relate this class to orbifold theory with high age. The question of birational invariance is investigated.

I classify the Gorenstein singularities of genus three. There are seven families, each corresponding to a connected component of a stratum of differentials.

The geometry of Gorenstein singularities appears to be intimately related to that of differentials, both from the algebraic and from the tropical point of view. A well-studied compactification of strata of differentials parametrises multi-scale differentials: in the limit, the curve C becomes nodal and reducible, and the differential may vanish along a subcurve; rescaling it, though, we obtain a non-zero meromorphic differential on this subcurve, possibly vanishing on a smaller subcurve, and so on. This compactification is in general not irreducible, but the main component can be described analytically in terms of the global residue condition (GRC). Here we state a conjectural algebraic characterisation: smoothable multiscale differentials are exactly those whose meromorphic parts descend to generate the dualising bundle of a Gorenstein contraction of C. As usual, producing the Gorenstein contraction is easy in one-parameter smoothing families but is tricky over a general base.  We address this issue in the case of logarithmic hyperelliptic curves (in the sense of admissible covers), by first contracting the tree of rational curves, and then reconstructing the Gorenstein contraction as a double cover. The combinatorial data needed is a realisable (i.e. coming from the tree) tropical differential on the subcurve to be contracted. Our construction is local around the subcurve and commutes with arbitrary base-change. As an application, we prove the above conjecture in the case of hyperelliptic (anti-invariant) differentials. 

Affine Bruhat--Tits buildings are geometric spaces extracting the combinatorics of algebraic groups. The building of PGL parametrizes flags of subspaces/lattices in or, equivalently, norms on a fixed finite-dimensional vector space, up to homothety. It has first been studied by Goldman and Iwahori as a piecewise-linear analogue of symmetric spaces. The space of seminorms compactifies the space of norms and admits a natural surjective restriction map from the Berkovich analytification of projective space. Inspired by Payne's result that analytification is the limit of all tropicalizations, we show that the space of seminorms is the limit of all tropicalized linear subspaces of rank r (as the embedding and the dimension of the ambient projective space vary), and prove a faithful tropicalization result for compactified linear spaces. The space of seminorms is in fact the tropicalization of the universal realizable valuated matroid, extending a result of Dress and Terhalle.

Several theories have been developed for the enumeration of curves tangent to a divisor, notably equipping both the curve and the target with an orbifold (root)/logarithmic structure. When the genus of the curves is zero and the divisor is smooth, these theories are known to agree [ACW, AMW]. When the genus is positive, the orbifold theory is polynomial in the rooting parameter(s): in the smooth divisor case, the constant term coincides with the relative theory [TY]. The same authors conjectured that the constant term would provide a route to the logarithmic theory in general [TY]. But root stacks are only one incarnation of logarithmic modifications, to which the logarithmic theory is known to be insensitive [AW], the other one being blowups of boundary strata. This stronger birational invariance needs to be forced upon the orbifold theory in order to match the logarithmic one. Concretely, for every fixed genus zero numerical data, we identify a condition, satisfied by any sufficiently refined blowup (Y,E)->(X,D), such that the logarithmic invariants of (X,D) equal the orbifold invariants of (Y,E).

Several theories have been developed for the enumeration of curves tangent to a divisor, notably equipping both the source and the target with an orbifold/logarithmic structure. When the genus of the curves is zero and the divisor is smooth, these theories are known to agree [ACW, AMW]. Moreover, in case the divisor is also nef, and we consider rational curves with only one point of (maximal) contact to it, the invariants agree with those of a "local" geometry, namely the total space of the dual line bundle associated to the divisor [vGGR]. What happens when the divisor is not smooth? Here we prove that the local and orbifold theories agree (with the naive one) in genus zero and maximal contact with respect to a sufficiently positive divisor with simple normal crossings, whereas in general the logarithmic theory is known to be different [NR] (although the numerical invariants often turn out to be the same, especially in the toric case [BBvG]).

As an application of our modular desingularisation, we study the moduli space of genus two (nodal) quartics in the projective plane. The moduli space of stable maps has more than twenty irreducible components, and for each of them we describe (pictorially as well!) its intersection with the main component. The smoothability criterion - generalising Vakil's in genus one - involves not only the singularities of the image curve, but also the node configuration on the source.

Moduli spaces of curves in projective space are arbitrarily singular by the Vakil-Murphy law. Yet, in 2006, Vakil and Zinger described an explicit sequence of blow-ups desingularising the main component of the space of stable maps in genus one. Here, we construct a smooth modular compactification of the space of genus two curves in projective space by extending the methods of Ranganathan, Santos-Parker and Wise (see this paper). Based on the author's observation that the geometry of Gorenstein singularities corresponds to that of the canonical linear system on their semistable models, we work over the space of admissible hyperelliptic covers, which, in genus two, encodes the Brill-Noether theory of nodal curves. With this logarithmic structure - which is richer than that of the moduli space of curves - the universal Gorenstein contraction can be constructed over a logarithmic modification of the base. Combinatorially, the latter is specified by the domain of linearity of a certain piecewise linear function on the tropicalisation of the universal nodal curve, which turns out to be a (realisable) tropical canonical divisor. Our second main finding is that not only isolated singularities, but also singular multiple curves (we call them "ribbons with tails") appear naturally in the universal Gorenstein family, both to account for the hyperelliptic component of the moduli space of maps, and to interpolate among various possible choices for the isolated singularities. Ultimately, we are able to define reduced Gromov-Witten invariants of genus two for any smooth projective target. It would be interesting to establish their relation with the standard invariants. Our approach complements that of Hu, Li and Niu.

We develop a reduced version of relative Gromov-Witten theory in genus one: we show that radially aligned maps satisfying a double factorisation condition provide a desingularisation of the main component of the (Kim's) moduli space. Then, we extend Gathmann's recursive scheme to this context: we unveil the tropical origin of his formula - which is of independent interest - and we use it to give an in-principle reduction of (i) the absolute reduced theory of the divisor, (ii) the relative reduced theory, and (iii) the rubber reduced theory to the absolute reduced theory of the ambient space. The most technically difficult part of the paper is the study of the interaction between the factorisation condition and the splitting principle for the boundary divisors of interest.

I introduce a series of alternative compactifications of the moduli space of smooth pointed curves of genus two. These are proper DM stacks containing the locus of smooth curves as a dense open, and contained in the stack of reduced Gorenstein curves. I classify all the isolated Gorenstein singularities of genus two, and provide a detailed description of their geometry, including crimping spaces and semistable tails. I highlight a fundamental relation with the geometry of the canonical system on nodal curves, both at an algebro-geometric and at a tropical level. The construction of these compactifications is inspired by Smyth's work, in particular by his notion of level. To stay in the Gorenstein locus, though, a hack is devised that exploits one of the markings as a reference point, thus breaking the usual symmetry.

A (mostly) expository paper on virtual intersection theory: we explain the relevant material from Fulton, Behrend-Fantechi, Manolache, Kim-Kresch-Pantev (etc.) through basic examples of scheme theory and enumerative geometry. At the end, we discuss some more advanced applications to Gromov-Witten theory.

We provide a modular interpretation of Zinger's reduced genus one Gromov-Witten invariants for the quintic threefold. Reduced invariants were introduced as a by-product of the Vakil-Zinger desingularisation of Kontsevich's moduli space of stable maps to the projective space in genus one. They satisfy quantum Lefschetz, so Zinger was able to compute them by torus localisation. Moreover, together with Jun Li, he proved a comparison formula with ordinary Gromov-Witten invariants, showing that the difference can be expressed in terms of genus zero invariants. Thus, he was able to give the first mathematical confirmation of the BCOV predictions from mirror symmetry. We equate the reduced invariants with those coming from Viscardi's alternative compactification, where cusps are allowed in the source curve  instead of elliptic tails.

Quasimaps provide a system of alternative compactifications of the space of morphisms from smooth curves to a large class of GIT quotients. Roughly speaking, rational tails are traded in for basepoints. We develop a theory of quasimaps tangent to a hypersurface, subject to the following restrictions: the source genus is zero, the ambient target variety is toric, and the divisor is a smooth hyperplane section. We extend Gathmann's recursive scheme for the computation of relative invariants, thus reproving some formulae of Ciocan-Fontanine and Kim from this paper. We put forward a relation with the relative I-function of Fan, Tseng, and You from this paper, providing a mirror symmetric motivation for our construction. The first arXiv version contains more foundational material on quasimaps.