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Francesca Carocci

Publications and source records attributed to Francesca Carocci.

15 recordsLinked to original sources
arXiv2026

A correlated refinement of the double double ramification cycle

Given a family of semi-stable curves together with two degree 0 line bundles, the double double ramification cycle measures the locus where both line bundles are trivial on the fibers. When the two line bundles come equipped with natural roots, we provide a refinement of the DDR-class using the Weil pairing of the roots. We prove that the refined classes satisfy a multiple cover formula analogous to the one for correlated invariants of projective bundles on elliptic curves proved in [BC25b]. As a consequence, we prove that log-GW invariants of toric surfaces can be refined taking into account the position of the points mapped to the toric boundary, and that these refined invariants also satisfy a multiple cover formula; the latter is as a variation of the N. Takahashi conjecture for genus zero maximal contact curves for P2 relative a smooth elliptic curve E.

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arXiv2026

Vector bundles on Olsson fans

Artin fans are algebro-geometric incarnations of cone complexes. We study weakly convex Olsson fans, generalising Artin fans in two ways: first, they admit lineality spaces, thus including tropical tori as well; second, they are defined over a base logarithmic scheme, thus providing a relative version of equivariant toric geometries. We determine conditions under which Olsson fans are well-behaved, in the sense that their geometry is determined combinatorially. We undertake the study of quasicoherent sheaves, and in particular vector bundles, on Olsson fans: we describe their moduli stack in the case of a (subdivided) cone, and some failures of algebraicity in the general case; Weyl convexity shows up naturally in this context.

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arXiv2025

Multiple cover formulas for abelian surfaces via correlated invariants

We prove the multiple cover formula conjecture for abelian surfaces for a large class of insertions, including all stationary invariants. The proof uses the reduced degeneration formula expressing the invariants in terms of the correlated Gromov--Witten invariants previously introduced by the authors.

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arXiv2025

Correlated double ramification cycle formula

We prove a refinement of Pixton's formula for the double ramification cycle with target variety which takes into account the correlator of a rubber map previously introduced by the authors. To do so, we need to: reinterpret the correlator in terms of (logarithmic) roots of the trivial line bundle; refine the natural stratification of the boundary of moduli of maps to keep track of how torsion line bundles on the target variety pull-back. We apply the refined DR cycle formula to compute 0-correlated invariants for genus g curve with points and a $\lambda$-class insertion.

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arXiv2024

Chow theory of toric variety bundles

We describe the Chow homology and cohomology of toric variety bundles, with no restrictions on the singularities of the fibre. We present the ordinary and equivariant homologies as modules over the cohomology of the base, identify the ordinary cohomology with homology-valued Minkowski weights, and identify the equivariant cohomology with cohomology-weighted piecewise polynomial functions. We describe the product structure on Minkowski weights via a fan displacement rule, and the non-equivariant limit via equivariant multiplicities. Along the way we establish relative analogues of the K\"unneth property and Kronecker duality. Applications include the balancing condition in logarithmic enumerative geometry.

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arXiv2024

Correlated Gromov-Witten Invariants

We introduce a geometric refinement of Gromov-Witten invariants for $\mathbb P^1$-bundles relative to the natural fiberwise boundary structure. We call these refined invariant correlated Gromov-Witten invariants. Furthermore, we prove a refinement of the degeneration formula keeping track of the correlation. Finally, combining certain invariance properties of the correlated invariant, a local computation and the refined degeneration formula we follow floor diagram techniques to prove regularity results for the generating series of the invariants in the case of $\mathbb P^1$-bundles over elliptic curves. Such invariants are expected to play a role in the degeneration formula for reduced Gromov-Witten invariants for abelian and K3 surfaces.

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arXiv2024

BPS invariants from $p$-adic integrals

We define $p$-adic BPS or $p$BPS-invariants for moduli spaces $M_{β,χ}$ of 1-dimensional sheaves on del Pezzo surfaces by means of integration over a non-archimedean local field $F$ . Our definition relies on a canonical measure $μ_{can}$ on the $F$-analytic manifold associated to $M_{β,χ}$ and the $p$BPS-invariants are integrals of natural $\mathbb{G}_m$-gerbes with respect to $μ_{can}$. A similar construction can be done for meromorphic Higgs bundles on a curve. Our main theorem is a $χ$-independence result for these $p$BPS-invariants. For 1-dimensional sheaves on del Pezzo surfaces and meromorphic Higgs bundles, we obtain as a corollary the agreement of $p$BPS with usual BPS-invariants trough a result of Maulik-Shen.

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arXiv2023

Rubber tori in the boundary of expanded stable maps

We investigate torus actions on logarithmic expansions in the context of enumerative geometry. Our main result is an intrinsic and coordinate-free description of the higher-rank rubber torus appearing in the boundary of the space of expanded stable maps. The rubber torus is constructed canonically from the tropical moduli space, and its action on each stratum of the expanded target is encoded in a linear tropical position map. The presence of 2-morphisms in the universal target forces expanded stable maps differing by the rubber action to be identified. This provides the first step towards a recursive description of the boundary of the expanded moduli space, with future applications including localisation and rubber calculus.

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arXiv2022

Tropical expansions and toric variety bundles

A tropical expansion is a degeneration of a toroidal embedding, induced by a polyhedral subdivision of its tropicalisation. Each irreducible component of a tropical expansion admits a collapsing map down to a stratum of the original variety. We study the relative geometry of this map. We give a polyhedral criterion for the map to have the structure of a toric variety bundle, and prove that this structure always exists over the interior of the codomain. We give examples demonstrating that this is the strongest statement one can hope for in general. In addition, we provide a combinatorial recipe for constructing the toric variety bundle as a fibrewise GIT quotient of an explicit split vector bundle. Our proofs make systematic use of Artin fans as a language for globalising local toric models.

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arXiv2021

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

We construct a modular desingularisation of $\overline{\mathcal{M}}_{2,n}(\mathbb{P}^r,d)^{\text{main}}$. The geometry of Gorenstein singularities of genus two leads us to consider maps from prestable admissible covers: with this enhanced logarithmic structure, it is possible to desingularise the main component by means of a logarithmic modification. Both isolated and non-reduced singularities appear naturally. Our construction gives rise to a notion of reduced Gromov-Witten invariants in genus two.

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arXiv2020

Virtual classes for the working mathematician

This note is intended to be a friendly introduction to virtual classes. We review virtual classes and we give a number of properties and applications. We also include a new virtual push-forward theorem and many computations of virtual classes in simple examples.

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arXiv2018

Endomorphisms of Koszul complexes: formality and application to deformation theory

We study the differential graded Lie algebra of endomorphisms of the Koszul resolution of a regular sequence on a unitary commutative $K$-algebra $R$ and we prove that it is homotopy abelian over $K$, while it is generally not formal over $R$. We apply this result to prove an annihilation theorem for obstructions of (derived) deformations of locally complete intersection ideal sheaves on projective schemes.

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arXiv2018

Reduced invariants from cuspidal maps

We consider genus 1 enumerative invariants arising from the Smyth-Viscardi moduli space of stable maps from curves with nodes and cusps. We prove that these invariants are equal to the Vakil-Zinger reduced invariants for the quintic threefold, providing a modular interpretation of the latter.

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arXiv2015

Homological projective duality for linear systems with base locus

We show how blowing up varieties in base loci of linear systems gives a procedure for creating new homological projective duals from old. Starting with a HP dual pair $X,Y$ and smooth orthogonal linear sections $X_L,Y_L$, we prove that the blowup of $X$ in $X_L$ is naturally HP dual to $Y_L$. The result does not need $Y$ to exist as a variety, i.e. it may be "noncommutative". We extend the result to the case where the base locus $X_L$ is a multiple of a smooth variety and the universal hyperplane has rational singularities; here the HP dual is a categorical resolution of singularities of $Y_L$. Finally we give examples where, starting with a noncommutative $Y$, the above process nevertheless gives geometric HP duals.

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