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Luca Schaffler

Publications and source records attributed to Luca Schaffler.

At least 19 recordsLinked to original sources
arXiv2025

Enriques surfaces with non-generic non-degeneracy

We study the non-degeneracy invariant $\mathrm{nd}(Y)$ of complex Enriques surfaces in families. Our first main result shows that $\mathrm{nd}(Y)$ cannot increase under specialization. The second main result is the conclusion of the computation of the non-degeneracy invariant for the $155$ families of $(\tau,\overline{\tau})$-generic surfaces introduced by Brandhorst and Shimada. Of the previously known $144$ cases, only $3$ satisfy $\mathrm{nd}(Y)\neq10$, which is the non-degeneracy invariant of a general Enriques surface. The remaining $11$ families studied in this article also have non-generic non-degeneracy. To compute this, we produce upper bounds on $\mathrm{nd}(Y)$ by refining this invariant into two others: the Fano and Mukai non-degeneracy invariants, which are related to two different classes of projective realizations of Enriques surfaces. As a result, we find the first known examples of Enriques surfaces with $\mathrm{nd}(Y)=9$.

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arXiv2025

Fineness and smoothness of a KSBA moduli of marked cubic surfaces

By work of Gallardo-Kerr-Schaffler, it is known that Naruki's compactification of the moduli space of marked cubic surfaces is isomorphic to the normalization of the Kollár, Shepherd-Barron, and Alexeev compactification parametrizing pairs $\left(S,\left(\frac{1}{9}+ε\right)D\right)$, with $D$ the sum of the $27$ marked lines on $S$, and their stable degenerations. In the current paper, we show that the normalization assumption is not necessary as we prove that this KSBA compactification is smooth. Additionally, we show it is a fine moduli space. This is done by studying the automorphisms and the $\mathbb{Q}$-Gorenstein obstructions of the stable pairs parametrized by it.

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arXiv2025

An explicit wall crossing for the moduli space of hyperplane arrangements

The moduli space of hyperplanes in projective space has a family of geometric and modular compactifications that parametrize stable hyperplane arrangements with respect to a weight vector. Among these, there is a toric compactification that generalizes the Losev-Manin moduli space of points on the line. We study the first natural wall crossing that modifies this compactification into a non-toric one by varying the weights. As an application of our work, we show that any $\mathbb{Q}$-factorialization of the blow up at the identity of the torus of the generalized Losev-Manin space is not a Mori dream space for a sufficiently high number of hyperplanes. Additionally, for lines in the plane, we provide a precise description of the wall crossing.

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arXiv2024

Simplices osculating rational normal curves

A classical result of von Staudt states that if eight planes osculate a twisted cubic curve and we divide them into two groups of four, then the eight vertices of the corresponding tetrahedra lie on a twisted cubic curve. In the current paper, we give an alternative proof of this result using modern tools, and at the same time we prove the analogous result for rational normal curves in any projective space. This higher dimensional generalization was claimed without proof in a paper of H.S. White in 1921.

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arXiv2024

The non-degeneracy invariant of Brandhorst and Shimada's families of Enriques surfaces

Brandhorst and Shimada described a large class of Enriques surfaces, called $(τ,\overlineτ)$-generic, for which they gave generators for the automorphism groups and calculated the elliptic fibrations and the smooth rational curves up to automorphisms. In the present paper, we give lower bounds for the non-degeneracy invariant of such Enriques surfaces, we show that in most cases the invariant has generic value $10$, and we present the first known example of complex Enriques surface with infinite automorphism group and non-degeneracy invariant not equal to $10$.

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arXiv2023

Gaussian likelihood geometry of projective varieties

We explore the maximum likelihood degree of a homogeneous polynomial $F$ on a projective variety $X$, $\mathrm{MLD}_F(X)$, which generalizes the concept of Gaussian maximum likelihood degree. We show that $\mathrm{MLD}_F(X)$ is equal to the count of critical points of a rational function on $X$, and give different geometric characterizations of it via topological Euler characteristic, dual varieties, and Chern classes.

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arXiv2023

Determinantal varieties from point configurations on hypersurfaces

We consider the scheme $X_{r,d,n}$ parametrizing $n$ ordered points in projective space $\mathbb{P}^r$ that lie on a common hypersurface of degree $d$. We show that this scheme has a determinantal structure and we prove that it is irreducible, Cohen-Macaulay, and normal. Moreover, we give an algebraic and geometric description of the singular locus of $X_{r,d,n}$ in terms of Castelnuovo-Mumford regularity and $d$-normality. This yields a characterization of the singular locus of $X_{2,d,n}$ and $X_{3,2,n}$.

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arXiv2023

Unimodal singularities and boundary divisors in the KSBA moduli of a class of Horikawa surfaces

Smooth minimal surfaces of general type with $K^2=1$, $p_g=2$, and $q=0$ constitute a fundamental example in the geography of algebraic surfaces, and the 28-dimensional moduli space $\mathbf{M}$ of their canonical models admits a modular compactification $\overline{\mathbf{M}}$ via the minimal model program. We describe eight new irreducible boundary divisors in such compactification parametrizing reducible stable surfaces. Additionally, we study the relation with the GIT compactification of $\mathbf{M}$ and the Hodge theory of the degenerate surfaces that the eight divisors parametrize.

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arXiv2022

A computational view on the non-degeneracy invariant for Enriques surfaces

For an Enriques surface $S$, the non-degeneracy invariant $\mathrm{nd}(S)$ retains information on the elliptic fibrations of $S$ and its polarizations. In the current paper, we introduce a combinatorial version of the non-degeneracy invariant which depends on $S$ together with a configuration of smooth rational curves, and gives a lower bound for $\mathrm{nd}(S)$. We provide a SageMath code that computes this combinatorial invariant and we apply it in several examples. First we identify a new family of nodal Enriques surfaces satisfying $\mathrm{nd}(S)=10$ which are not general and with infinite automorphism group. We obtain lower bounds on $\mathrm{nd}(S)$ for the Enriques surfaces with eight disjoint smooth rational curves studied by Mendes Lopes-Pardini. Finally, we recover Dolgachev and Kondō's computation of the non-degeneracy invariant of the Enriques surfaces with finite automorphism group and provide additional information on the geometry of their elliptic fibrations.

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arXiv2022

Decomposition of Lagrangian classes on K3 surfaces

We study the decomposability of a Lagrangian homology class on a K3 surface into a sum of classes represented by special Lagrangian submanifolds, and develop criteria for it in terms of lattice theory. As a result, we prove the decomposability on an arbitrary K3 surface with respect to the Kähler classes in dense subsets of the Kähler cone. Using the same technique, we show that the Kähler classes on a K3 surface which admit a special Lagrangian fibration form a dense subset also. This implies that there are infinitely many special Lagrangian 3-tori in any log Calabi-Yau 3-fold.

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arXiv2022

Families of pointed toric varieties and degenerations

The Losev-Manin moduli space parametrizes pointed chains of projective lines. In this paper we study a possible generalization to families of pointed degenerate toric varieties. Geometric properties of these families, such as flatness and reducedness of the fibers, are explored via a combinatorial characterization. We show that such families are described by a specific type of polytope fibration which generalizes the twisted Cayley sums, originally introduced to characterize elementary extremal contractions of fiber type associated to projective $\mathbb{Q}$-factorial toric varieties with positive dual defect. The case of a one-dimensional simplex can be viewed as an alternative construction of the permutohedra.

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arXiv2021

The KSBA compactification of the moduli space of $D_{1,6}$-polarized Enriques surfaces

We describe a compactification by stable pairs (also known as KSBA compactification) of the $4$-dimensional family of Enriques surfaces which arise as the $\mathbb{Z}_2^2$-covers of the blow up of $\mathbb{P}^2$ at three general points branched along a configuration of three pairs of lines. Up to a finite group action, we show that this compactification is isomorphic to the toric variety associated to the secondary polytope of the unit cube. We relate the KSBA compactification considered to the Baily-Borel compactification of the same family of Enriques surfaces. Part of the KSBA boundary has a toroidal behavior, another part is isomorphic to the Baily-Borel compactification, and what remains is a mixture of these two. We relate the stable pair compactification studied here with Looijenga's semitoric compactifications.

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arXiv2021

Compactifications of moduli of points and lines in the projective plane

Projective duality identifies the moduli spaces $\mathbf{B}_n$ and $\mathbf{X}(3,n)$ parametrizing linearly general configurations of $n$ points in $\mathbb{P}^2$ and $n$ lines in the dual $\mathbb{P}^2$, respectively. The space $\mathbf{X}(3,n)$ admits Kapranov's Chow quotient compactification $\overline{\mathbf{X}}(3,n)$, studied also by Lafforgue, Hacking, Keel, Tevelev, and Alexeev, which gives an example of a KSBA moduli space of stable surfaces: it carries a family of certain reducible degenerations of $\mathbb{P}^2$ with $n$ "broken lines". Gerritzen and Piwek proposed a dual perspective, a compact moduli space parametrizing certain reducible degenerations of $\mathbb{P}^2$ with $n$ smooth points. We investigate the relation between these approaches, answering a question of Kapranov from 2003.

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arXiv2021

A Pascal's Theorem for rational normal curves

Pascal's Theorem gives a synthetic geometric condition for six points $a,\ldots,f$ in $\mathbb{P}^2$ to lie on a conic. Namely, that the intersection points $\overline{ab}\cap\overline{de}$, $\overline{af}\cap\overline{dc}$, $\overline{ef}\cap\overline{bc}$ are aligned. One could ask an analogous question in higher dimension: is there a coordinate-free condition for $d+4$ points in $\mathbb{P}^d$ to lie on a degree $d$ rational normal curve? In this paper we find many of these conditions by writing in the Grassmann-Cayley algebra the defining equations of the parameter space of $d+4$ ordered points in $\mathbb{P}^d$ that lie on a rational normal curve. These equations were introduced and studied in a previous joint work of the authors with Giansiracusa and Moon. We conclude with an application in the case of seven points on a twisted cubic.

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arXiv2021

KSBA compactification of the moduli space of K3 surfaces with purely non-symplectic automorphism of order four

We describe a compactification by KSBA stable pairs of the five-dimensional moduli space of K3 surfaces with purely non-symplectic automorphism of order four and $U(2)\oplus D_4^{\oplus2}$ lattice polarization. These K3 surfaces can be realized as the minimal resolution of the double cover of $\mathbb{P}^1\times\mathbb{P}^1$ branched along a specific $(4,4)$ curve. We show that, up to a finite group action, this stable pair compactification is isomorphic to Kirwan's partial desingularization of the GIT quotient $(\mathbb{P}^1)^8//\mathrm{SL}_2$ with the symmetric linearization.

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arXiv2021

Point configurations, phylogenetic trees, and dissimilarity vectors

In 2004 Pachter and Speyer introduced the higher dissimilarity maps for phylogenetic trees and asked two important questions about their relation to the tropical Grassmannian. Multiple authors, using independent methods, answered affirmatively the first of these questions, showing that dissimilarity vectors lie on the tropical Grassmannian, but the second question, whether the set of dissimilarity vectors forms a tropical subvariety, remained opened. We resolve this question by showing that the tropical balancing condition fails. However, by replacing the definition of the dissimilarity map with a weighted variant, we show that weighted dissimilarity vectors form a tropical subvariety of the tropical Grassmannian in exactly the way that Pachter--Speyer envisioned. Moreover, we provide a geometric interpretation in terms of configurations of points on rational normal curves and construct a finite tropical basis that yields an explicit characterization of weighted dissimilarity vectors.

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arXiv2020

Geometric interpretation of toroidal compactifications of moduli of points in the line and cubic surfaces

It is known that some GIT compactifications associated to moduli spaces of either points in the projective line or cubic surfaces are isomorphic to Baily-Borel compactifications of appropriate ball quotients. In this paper, we show that their respective toroidal compactifications are isomorphic to moduli spaces of stable pairs as defined in the context of the MMP. Moreover, we give a precise mixed-Hodge-theoretic interpretation of this isomorphism for the case of eight labeled points in the projective line.

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