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Stefan Schröer

Publications and source records attributed to Stefan Schröer.

At least 19 recordsLinked to original sources
arXiv2026

Surfaces of general type and sl_2-triples

The sl_2-triples play a fundamental role for the structure theory of Lie algebras, and representation theory in general. Here we investigate sl_2-triples of global vector fields on schemes X in positive characteristics p>0, and develop a general theory for actions of the corresponding height-one group scheme G=SL_2[F]. Sending a point to the Lie algebra of its stabilizer defines rational maps to various Grassmann varieties. For surfaces of general type, this yields fibrations in curves of genus g at least 2 over the projective line. Using properties of the corresponding moduli stack M_g, we prove that there are no smooth surfaces of general type with an sl_2-triple. On the other hand, employing Lefschetz pencils and Frobenius pullbacks we show that canonical surfaces of general type with such triples exist in abundance. In this connection, we classify the rational double points where the tangent sheaf is free or the evaluation pairing with Kähler differentials is surjetive, including characteristic two.

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arXiv2026

K3 surfaces over small number fields and Kummer constructions in families

We construct K3 surfaces over number fields that have good reduction everywhere. These do not exists over the rational numbers, by results of Abrashkin and Fontaine. Our surfaces exist for three quadratic number fields, and an infinite family of $S_3$-number fields. To this end we develop a theory of Kummer constructions in families, based on Romagny's notion of the effective models, here applied to sign involutions. This includes quotients of non-normal surfaces by infinitesimal group schemes in characteristic two, as developed by Kondo and myself. By the results of Brieskorn and Artin, the resulting families of normal K3 surfaces admit simultaneous resolutions of singularities, at least after suitable base-changes. These resolutions are constructed in two ways: First, by blowing-up families of one-dimensional centers that acquire embedded components. Second, by computing various l-adic local systems in terms of representation theory, and invoking Shepherd-Barrons results on the resolution functor, which is representable by a highly non-separated algebraic space.

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arXiv2026

A simple proof for Hochster's Theorem

We give a conceptual proof for Hochster's Theorem, which asserts that each spectral space is homeomorphic to the spectrum of a ring. Given a ground field and a spectral space, our ring is constructed as filtered direct limit of prime-finite ring, which are attached in a functorial way to finite Kolmogoroff spaces. The construction simplifies an argument of Ershov along these lines. Our crucial ingredient is an assembly of finite Kolmogoroff spaces in terms of coequalizers and pushouts of one-dimensional spaces, and Schwede's observation on prime ideals in cartesian squares of rings.

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arXiv2026

Deligne-Lusztig varieties whose canonical divisors have negativity

We investigate compactified Deligne-Lusztig varieties whose canonical divisor, when expressed as a linear combination of boundary divisors, has all coefficients strictly negative or zero. In dimension two we obtain explicit descriptions: The case $C_2$ gives the supersingular K3 surface with Artin invariant $σ=1$ in characteristic two. The arguments rely on properties of the Tutte-Coxeter graph; in this connection we also gain some insight into the arithmetic of quasi-elliptic Weierstrass equations and rational double points. The twisted cases ${}^2A_2$ and ${}^2C_2$ and ${}^2G_2$ yield particular ruled surfaces, attached in a canonical way to supersingular elliptic curves in characteristic two, or the Ree curve in characteristic three. The latter is a curve of genus fifteen with outstanding symmetries. In the former cases, the surfaces can also be expressed as symmetric squares. To obtain such results, we develop a new general framework for Deligne-Lusztig varieties that relies on the Isogeny Theorem, and works over prime fields rather than their algebraic closures, and includes the notorious Suzuki-Ree cases without effort.

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arXiv2025

Varieties with free tangent sheaves

We coin the term \emph{$T$-trivial varieties} to denote smooth proper schemes over ground fields $k$ whose tangent sheaf is free. Over the complex numbers, this are precisely the abelian varieties. However, Igusa observed that in characteristic $p\leq 3$ certain bielliptic surfaces are $T$-trivial. We show that $T$-trivial varieties $X$ separably dominated by abelian varieties $A$ can exist only for $p\leq 3$. Furthermore, we prove that every $T$-trivial variety, after passing to a finite étale covering, is fibered in $T$-trivial varieties with Betti number $b_1=0$. We also show that if some $n$-dimensional $T$-trivial $X$ lifts to characteristic zero and $p\geq 2n+2$ holds, it admits a finite étale covering by an abelian variety. Along the way, we establish several results about the automorphism group of abelian varieties, and the existence of relative Albanese maps.

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arXiv2025

The Iskovskih Theorem for regular surfaces over imperfect fields

We generalize Iskovskih's theorem about surfaces without irregularity and bigenus from the smooth case to regular surfaces over arbitrary fields, with special focus on the case of imperfect fields. This includes surfaces that are geometrically non-normal or geometrically non-reduced. Here the usual approach of Galois descent breaks down, and one relies entirely on the scheme theory over the ground field. Moreover, the degrees of closed points can be larger than expected, and certain curves might have purely inseparable constant field extension. To deal with the latter we establish a general theory for inseparable pencils, which is of independent interest. A crucial case not present in the classical proof for Iskovskih's theorem leads to non-normal quartic surfaces that are singular along a twisted cubic, or more exotic space curves of degree three.

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arXiv2025

Rational points in coarse moduli spaces and twisted representations

We study moduli spaces and moduli stacks for representations of associative algebras in Azumaya algebras, in rather general settings. We do not impose any stability condition and work over arbitrary ground rings, but restrict attention to the so-called Schur representations, where the only automorphisms are scalar multiplications. The stack comprises twisted representations, which are representations that live on the gerbe of splittings for the Azumaya algebra. Such generalized spaces and stacks appear naturally: For any rational point on the classical coarse moduli space of matrix representations, the machinery of non-abelian cohomology produces a modified moduli problem for which the point acquires geometric origin. The latter are given by representations in Azumaya algebras.

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arXiv2024

The inverse Galois problem for connected algebraic groups

We show that each connected group scheme of finite type over an arbitrary ground field is isomorphic to the component of the identity inside the automorphism group scheme of some projective, geometrically integral scheme. The main ingredients are embeddings into smooth group schemes, equivariant completions, blow-ups of orbit closures, Fitting ideals for Kähler differentials, and Blanchard's Lemma.

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arXiv2024

Sign involutions on para-abelian varieties

We study the so-called sign involutions on twisted forms of abelian varieties, and show that such a sign involution exists if and only if the class in the Weil--Châtelet group is annihilated by two. If these equivalent conditions hold, we prove that the Picard scheme of the quotient is étale and contains no points of finite order. In dimension one, such quotients are Brauer--Severi curves, and we analyze the ensuing embeddings of the genus-one curve into twisted forms of Hirzebruch surfaces and weighted projective spaces.

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arXiv2024

Loops on schemes and the algebraic fundamental group

In this note we give a re-interpretation of the algebraic fundamental group for proper schemes that is rather close to the original definition of the fundamental group for topological spaces. The idea is to replace the standard interval from topology by what we call interval schemes. This leads to an algebraic version of continuous loops, and the homotopy relation is defined in terms of the monodromy action. Our main results hinge on Macaulayfication for proper schemes and Lefschetz type results.

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arXiv2024

Generalizations of quasielliptic curves

We generalize the notion of quasielliptic curves, which have infinitesimal symmetries and exist only in characteristic two and three, to a remarkable hierarchy of regular curves having infinitesimal symmetries, defined in all characteristics and having higher genera. This relies on the study of certain infinitesimal group schemes acting on the affine line and certain compactifications. The group schemes are defined in terms of invertible additive polynomials over rings with nilpotent elements, and the compactification is constructed with the theory of numerical semigroups. The existence of regular twisted forms relies on Brion's recent theory of equivariant normalization. Furthermore, extending results of Serre from the realm of group cohomology, we describe non-abelian cohomology for semidirect products, to compute in special cases the collection of all twisted forms.

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arXiv2023

Para-abelian varieties and Albanese maps

We construct for every proper algebraic space over a ground field an Albanese map to a para-abelian variety, which is unique up to unique isomorphism. This holds in the absence of rational points or ample sheaves, and also for reducible or non-reduced spaces, under the mere assumption that the structure morphism is in Stein factorization. It also works under suitable assumptions in families. In fact the treatment of the relative setting is crucial, even to understand the situation over ground fields. This also ensures that Albanese maps are equivariant with respect to actions of group schemes. Our approach depends on the notion of families of para-abelian varieties, where each geometric fiber admits the structure of an abelian variety, and representability of tau-parts in relative Picard groups, together with structure results on algebraic groups.

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arXiv2023

Albanese maps for open algebraic spaces

We show that for each algebraic space that is separated and of finite type over a field, and whose affinization is connected and reduced, there is a universal morphism to a para-abelian variety. The latter are schemes that acquire the structure of an abelian variety after some ground field extension. This generalizes classical results of Serre on universal morphisms from algebraic varieties to abelian varieties. Our proof relies on corresponding facts for the proper case, together with the structural properties of group schemes, removal of singularities by alterations, and ind-objects. It turns out that the formation of the Albanese variety commutes with base-change up to universal homeomorphisms. We also give a detailed analysis of Albanese maps for algebraic curves and algebraic groups, with special emphasis on imperfect ground fields.

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arXiv2022

The structure of regular genus-one curves over imperfect fields

Working over imperfect fields, we give a comprehensive classification of genus-one curves that are regular but not geometrically regular, extending the known case of geometrically reduced curves. The description is given intrinsically, in terms of twisted forms of standard models with respect to infinitesimal group scheme actions, and not via extrinsic equations. The main new idea is to analyze and exploit moduli for fields of representatives in Cohen's Structure Theorem. The results serve for the understanding of genus-one fibrations over higher-dimensional bases.

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arXiv2022

The structure of Frobenius kernels for automorphism group schemes

We establish structure results for Frobenius kernels of automorphism group schemes for surfaces of general type in positive characteristics. It turns out that there are surprisingly few possibilities. This relies on properties of the famous Witt algebra, which is a simple Lie algebra without finite-dimensional counterpart over the complex numbers, together with is twisted forms. The result actually holds true for arbitrary proper integral schemes under the assumption that the Frobenius kernel has large isotropy group at the generic point. This property is measured by a new numerical invariant called the foliation rank.

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arXiv2022

There is no Enriques surface over the integers

We show that there is no family of Enriques surfaces over the ring of integers. This extends non-existence results of Minkowski for families of finite étale schemes, of Tate and Ogg for families of elliptic curves, and of Fontaine and Abrashkin for families of abelian varieties and more general smooth proper schemes with certain restrictions on Hodge numbers. Our main idea is to study the local system of numerical classes of invertible sheaves. Among other things, our result also hinges on counting rational points, Lang's classification of rational elliptic surfaces in characteristic two, the theory of exceptional Enriques surfaces due to Ekedahl and Shepherd-Barron, some recent results on the base of their versal deformation, Shioda's theory of Mordell--Weil lattices, and an extensive combinatorial study for the pairwise interaction of genus-one fibrations.

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arXiv2022

The resolution property holds away from codimension three

The purpose of this paper is to verify a conjecture of Gross under mild hypothesis: all reduced, separated, and excellent schemes have the resolution property away from a closed subset of codimension at least three. Our technique uses formal-local descent and the existence of affine flat neighborhoods to reduce the problem to constructing certain modules over commutative rings. Once in the category of modules we exhibit enough locally free sheaves directly, thereby establishing the resolution property for a specific class of algebraic spaces. A crucial step is showing it suffices to resolve a single coherent sheaf.

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