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Anne-Sophie Kaloghiros

Publications and source records attributed to Anne-Sophie Kaloghiros.

15 recordsLinked to original sources
arXiv2026

Degenerations of elliptic quartics and K-moduli of Fano threefolds

We study the K-moduli space of Fano threefolds obtained by first blowing up $\mathbb{P}^3$ along an elliptic quartic curve $C$ and then blowing up a fiber $\ell_p$ of the exceptional divisor $E \to C$. We prove that this K-moduli space is isomorphic to a VGIT quotient parametrizing pairs $(C,p)$, with linearization induced by the CM line bundle. In particular, we classify all K-(semi/poly)stable members of this deformation family. The main new ingredients in the proof include the geometry of the Hilbert scheme of elliptic quartic curves, deformation theory of Fano--K3 pairs, and optimal bounds on the local volumes of threefold singularities.

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arXiv2026

The boundary of K-moduli of prime Fano threefolds of genus twelve

We study the K-moduli stack of prime Fano threefolds of genus twelve, known as $V_{22}$. We prove that its boundary, which parametrizes singular members, is purely divisorial and consists of four irreducible components corresponding to the four families of Prokhorov's one-nodal $V_{22}$. A key ingredient is a modular relation between Fano threefolds $X$ and their anticanonical K3 surfaces $S$. We prove that the forgetful morphism from the moduli of Fano--K3 pairs $(X,S)$ where $X$ is a K-semistable degeneration of $V_{22}$ to the moduli space of genus $12$ polarized K3 surfaces $(S,{-K_X}|_S)$ is an open immersion. In particular, the K-moduli of $V_{22}$ is governed by the moduli of their anticanonical K3 surfaces, providing a modular realization of Mukai's philosophy. Along the way, we develop a general deformation framework for Fano threefolds of large volume, which may be useful beyond the study of K-moduli.

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arXiv2024

K-moduli of pure states of four qubits

We find all K-polystable limits of divisors in $(\mathbb{P}^1)^4$ of degree $(1,1,1,1)$ and explicitly describe the associated irreducible component of the K-moduli space.

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arXiv2021

On toric geometry and K-stability of Fano varieties

We present some applications of the deformation theory of toric Fano varieties to K-(semi/poly)stability of Fano varieties. First, we present two examples of K-polystable toric Fano 3-fold with obstructed deformations. In one case, the K-moduli spaces and stacks are reducible near the closed point associated to the toric Fano 3-fold, while in the other they are non-reduced near the closed point associated to the toric Fano 3-fold. Second, we study K-stability of the general members of two deformation families of smooth Fano 3-folds by building degenerations to K-polystable toric Fano 3-folds.

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arXiv2015

Non-rigid quartic 3-folds

Let $X\subset \mathbb{P}^4$ be a terminal factorial quartic $3$-fold. If $X$ is non-singular, $X$ is \emph{birationally rigid}, i.e. the classical MMP on any terminal $\mathbb{Q}$-factorial projective variety $Z$ birational to $X$ always terminates with $X$. This no longer holds when $X$ is singular, but very few examples of non-rigid factorial quartics are known. In this article, we first bound the local analytic type of singularities that may occur on a terminal factorial quartic hypersurface $X\subset \mathbb{P}^4$. A singular point on such a hypersurface is either of type $cA_n$ ($n\geq 1$), or of type $cD_m$ ($m\geq 4$), or of type $cE_6, cE_7$ or $cE_8$. We first show that if $(P \in X)$ is of type $cA_n$, $n$ is at most $7$, and if $(P \in X)$ is of type $cD_m$, $m$ is at most $8$. We then construct examples of non-rigid factorial quartic hypersurfaces whose singular loci consist (a) of a single point of type $cA_n$ for $2\leq n\leq 7$ (b) of a single point of type $cD_m$ for $m= 4$ or $5$ and (c) of a single point of type $cE_k$ for $k=6,7$ or $8$.

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arXiv2015

The Sarkisov program for Mori fibred Calabi-Yau pairs

We prove a version of the Sarkisov program for volume preserving birational maps of Mori fibred Calabi-Yau pairs valid in all dimensions. Our theorem generalises the theorem of Usnich and Blanc on factorisations of birational maps of the 2-dimensional torus that preserve the canonical volume form.

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arXiv2012

Finite generation and geography of models

There are two main examples where a version of the Minimal Model Program can, at least conjecturally, be performed successfully: the first is the classical MMP associated to the canonical divisor, and the other is Mori Dream Spaces. In this paper we formulate a framework which generalises both of these examples. Starting from divisorial rings which are finitely generated, we determine precisely when we can run the MMP, and we show why finite generation alone is not sufficient to make the MMP work.

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arXiv2012

Relations in the Sarkisov Program

The Sarkisov Program studies birational maps between varieties that are end products of the Minimal Model Program (MMP) on nonsingular uniruled varieties. If X and Y are terminal Q-factorial projective varieties endowed with a structure of Mori fibre space, any birational map between them can be decomposed into a finite number of elementary Sarkisov links. This decomposition is not unique in general, and any two distinct decompositions define a relation in the Sarkisov Program. This paper shows that relations in the Sarkisov Program are generated by some elementary relations. Roughly speaking, elementary relations are the relations among the end products of the MMP of Z over W, for suitable Z and W with relative Picard rank 3.

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arXiv2009

The defect of Fano 3-folds

This paper studies the defect of terminal Gorenstein Fano 3 folds. I determine a bound on the defect of terminal Gorenstein Fano 3-folds of Picard rank 1 that do not contain a plane. I give a general bound for quartic 3-folds and indicate how to study the defect of terminal Gorenstein Fano 3-folds with Picard rank 1 that contain a plane.

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arXiv2009

A classification of terminal quartic 3-folds and applications to rationality questions

This paper studies the birational geometry of terminal Gorenstein Fano 3-folds. If Y is not Q-factorial, in most cases, it is possible to describe explicitly the divisor class group Cl Y by running a Minimal Model Program (MMP) on X, a small Q-factorialisation of Y. In this case, Weil non-Cartier divisors are generated by "topological traces " of K-negative extremal contractions on X. One can show, as an application of these methods, that a number of families of non-factorial terminal Gorenstein Fano 3-folds are rational. In particular, I give some examples of rational quartic hypersurfaces with Cl Y of rank 2, and show that when Cl Y has rank greater than 6, Y is always rational.

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arXiv2008

Introduction to the Minimal Model Program and the existence of flips

The first aim of this note is to give a concise, but complete and self-contained, presentation of the fundamental theorems of Mori theory - the nonvanishing, base point free, rationality and cone theorems - using modern methods of multiplier ideals, Nadel vanishing, and the subadjunction theorem of Kawamata. The second aim is to write up a complete, detailed proof of existence of flips in dimension n assuming the minimal model program with scaling in dimension n-1.

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arXiv2007

The topology of terminal quartic 3-folds

In this thesis, I determine a bound on the defect of terminal Gorenstein quartic 3-folds. More generally, I study the defect of terminal Gorenstein Fano 3-folds of Picard rank 1 and genus at least 3. I state a geometric "motivation" of non Q-factoriality in the case of quartics.

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