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Camilla Felisetti

Publications and source records attributed to Camilla Felisetti.

11 recordsLinked to original sources
arXiv2026

Parabolic bundles and the intersection cohomology of moduli spaces of vector bundles on curves

The study of the intersection cohomology of moduli spaces of semistable bundles was initiated by Frances Kirwan in the 1980s. In this paper, we develop a self-contained, purely geometric framework to study the intersection cohomology of the moduli spaces of semistable degree-0 vector bundles of arbitrary rank on Riemann surfaces. Motivated by the 2015 work of Mozgovoy and Reineke, our approach applies the Beilinson-Bernstein-Deligne-Gabber Decomposition Theorem to the forgetful map from a parabolic moduli space. We give a detailed description of the topology of this map, the geometry of its fibers, and the precise structure of the relevant local systems, making systematic use of the multiplicative structures of the theory. As a global application of this machinery, we obtain a geometric proof of a recursive formula that reduces the calculation of the intersection Betti numbers of the degree-0 moduli spaces to the known formulas for the smooth, degree-1 moduli spaces.

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arXiv2026

Intersection theory on singular moduli spaces of vector bundles: a parabolic approach

We present explicit formulas for the intersection pairing in the intersection cohomology of the moduli space $M_0(r)$ of rank-$r$, degree-$0$ semistable bundles on a Riemann surface. The key idea is to realize this intersection cohomology as a canonical subspace of the cohomology of a smooth moduli space of parabolic bundles, where the pairing can be computed via the Hecke correspondence and the Jeffrey-Kirwan iterated residue formulas. This approach provides a simpler alternative to the blow-up construction of Jeffrey-Kirwan-Kiem-Woolf, yielding formulas for the intersection pairing on $M_0(r)$, for arbitrary $r$, with a clear geometric interpretation.

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arXiv2024

$P=W$ phenomena in algebraic and enumerative geometry

In view of the recent proofs of the P=W conjecture, the present paper reviews and relates the latest results in the field, with a view on how P=W phenomena appear in multiple areas of algebraic geometry. As an application, we give a detailed sketch of the proof of P=W by Maulik, Shen and Yin.

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arXiv2024

O'Grady tenfolds as moduli spaces of sheaves

We give a lattice-theoretic characterization for a manifold of $\mathrm{OG}10$ type to be birational to some moduli space of (twisted) sheaves on a K3 surface. We apply it to the Li-Pertusi-Zhao variety of $\mathrm{OG}10$ type associated to any smooth cubic fourfold. Moreover we determine when a birational transformation is induced by an automorphism of the K3 surface and we use this to classify all induced birational symplectic involutions.

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arXiv2022

P=W conjectures for character varieties with symplectic resolution

We establish P=W and PI=WI conjectures for character varieties with structural group $\mathrm{GL}_n$ and $\mathrm{SL}_n$ which admit a symplectic resolution, i.e. for genus 1 and arbitrary rank, and genus 2 and rank 2. We formulate the P=W conjecture for resolution, and prove it for symplectic resolutions. We exploit the topology of birational and quasi-étale modifications of Dolbeault moduli spaces of Higgs bundles. To this end, we prove auxiliary results of independent interest, like the construction of a relative compactification of the Hodge moduli space for reductive algebraic groups, and the projectivity of the compactification of the de Rham moduli space. In particular, we study in detail a Dolbeault moduli space which is specialization of the singular irreducible holomorphic symplectic variety of type O'Grady 6.

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arXiv2021

On the Splitting Principle of Beniamino Segre

We state and prove in modern terms a Splitting Principle first claimed by Beniamino Segre in 1938, which should be regarded as a strong form of the classical Principle of Connectedness.

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arXiv2021

Betti numbers of Brill-Noether varieties on a general curve

We compute the rational cohomology groups of the smooth Brill-Noether varieties $G^r_d(C)$, parametrizing linear series of degree $d$ and dimension exactly $r$ on a general curve $C$. As an application, we determine the whole intersection cohomology of the singular Brill-Noether loci $W^r_d(C)$, parametrizing complete linear series on $C$ of degree $d$ and dimension at least $r$.

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arXiv2021

On intersection cohomology and Lagrangian fibrations of irreducible symplectic varieties

We prove several results concerning the intersection cohomology and the perverse filtration associated with a Lagrangian fibration of an irreducible symplectic variety. We first show that the perverse numbers only depend on the deformation equivalence class of the ambient variety. Then we compute the border of the perverse diamond, which further yields a complete description of the intersection cohomology of the Lagrangian base and the invariant cohomology classes of the fibers. Lastly, we identify the perverse and Hodge numbers of intersection cohomology when the irreducible symplectic variety admits a symplectic resolution. These results generalize some earlier work by the second and third authors in the nonsingular case.

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arXiv2021

Intersection cohomology of the moduli space of Higgs bundles on a genus 2 curve

Let $C$ be a smooth projective curve of genus $2$. Following a method by O' Grady, we construct a semismall desingularization $\tilde{\mathcal{M}}_{Dol}^G$ of the moduli space $\mathcal{M}_{Dol}^G$ of semistable $G$-Higgs bundles of degree 0 for $G=GL(2,\mathbb{C}), SL(2,\mathbb{C})$. By the decomposition theorem by Beilinson, Bernstein, Deligne one can write the cohomology of $\tilde{\mathcal{M}}_{Dol}^G$ as a direct sum of the intersection cohomology of $\mathcal{M}_{Dol}^G$ plus other summands supported on the singular locus. We use this splitting to compute the intersection cohomology of $\mathcal{M}_{Dol}^G$ and prove that the mixed Hodge structure on it is actually pure, in analogy with what happens to ordinary cohomology in the smooth case of coprime rank and degree.

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arXiv2021

A support theorem for nested Hilbert schemes of planar curves

Consider a family of integral complex locally planar curves. We show that under some assumptions on the basis, the relative nested Hilbert scheme is smooth. In this case, the decomposition theorem of Beilinson, Bernstein and Deligne asserts that the pushforward of the constant sheaf on the relative nested Hilbert scheme splits as a direct sum of shifted semisimple perverse sheaves. We will show that no summand is supported in positive codimension.

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