arXiv subjects
Publications and source records attributed to Frank-Olaf Schreyer.
A general algebraic surface cannot be embedded in P4. Proving a conjecture by Hartshorne and Lichtenbaum, Ellingsrud and Peskine showed that there is a degree bound for smooth rational surfaces in P4, and in fact for surfaces not of general type. We give a survey of the classification status and of classical and computer-aided constructions of smooth non-general type surfaces in P4.
Let p in X be a point on a compact Riemann surface. The Weierstrass semigroup of p is the semigroup of pole orders of meromorphic functions on X that are regular at all but p. Hurwitz asked in 1892 whether all numerical semigroups occur as Weierstrass semigroups. In this paper we give a new method for showing that certain numerical semigroups are not Weierstrass, including some of every genus g in which non-Weierstrass examples could possibly exist, except g=18. Our example for g=13 has at once the smallest possible genus, multiplicity and number of generators of any possible non-Weierstrass semigroup.
Using the connection between hyperelliptic curves, Clifford algebras, and complete intersections $X$ of two quadrics, we describe Ulrich bundles on $X$ and construct some of minimal possible rank.
We give illustrative examples of how the computer algebra system OSCAR can support research in commutative algebra and algebraic geometry. We start with a thorough introduction to Groebner basis techniques, with particular emphasis on the computation of syzygies, then apply these techniques to deal with ideal and ring theoretic concepts such as primary decomposition and normalization, and finally use them for geometric case studies which concern curves and surfaces, both from a local and global point of view.
We discuss the space $VPS(Q,H)$ of ideals with Hilbert function $H=(1,n,n, \ldots)$ that are apolar to a full rank quadric $Q$. We prove that its components of saturated ideals are closely related to the locus of Gorenstein algebras and to the Slip component in border apolarity. We also point out an important error in~[RS] and provide the necessary corrections.
In this note we give a computationally easy to use method to compute a maximal extension of certain varieties. As a application we prove that a general paracanonical curve C genus 6 as a codimension three subvarieties of P^4 extend to precisely 26 families of surfaces Y in P^5.
Few explicit families of 3-folds are known for which the computation of the canonical ring is accessible and the birational geometry non-trivial. In this note we investigate a family of determinantal 3-folds in $\mathbb P^2 \times \mathbb P^3$ where this is the case.
In this paper we study marked numerical Godeaux surfaces with special bicanonical fibers. Based on our construction method of marked Godeaux surfaces we give a complete characterization for the existence of hyperelliptic bicanonical fibers and torsion fibers. Moreover, we describe how the families of Reid and Miyaoka with torsion $\mathbb{Z}/3\mathbb{Z}$ and $\mathbb{Z}/5\mathbb{Z}$ arise in our homological setting.
In this paper we describe a construction method for numerical Godeaux surfaces based on homological algebra. We show the existence of an 8-dimensional locally complete family of simply connected numerical Godeaux surfaces.
We show that the strata $ \mathcal{M}_{11,6}(k) \subset \mathcal{M}_{11} $ of $ 6-$gonal curves of genus $ 11 $, equipped with $k$ mutually independent and type I pencils of degree six, have a unirational irreducible component for $5\leq k\leq 9$. The unirational families arise from degree $ 9 $ plane curves with $ 4 $ ordinary triple and $ 5 $ ordinary double points that dominate an irreducible component of expected dimension. We will further show that the family of degree $ 8 $ plane curves with $ 10 $ ordinary double points covers an irreducible component of excess dimension in $ \mathcal{M}_{11,6}(10) $.
We study the variety membership testing problem in the case when the variety is given as an orbit closure and the ambient space is the set of all 3-tensors. The first variety that we consider is the slice rank variety, which consists of all 3-tensors of slice rank at most $r$. We show that the membership testing problem for the slice rank variety is $\NP$-hard. While the slice rank variety is a union of orbit closures, we define another variety, the minrank variety, expressible as a single orbit closure. Our next result is the $\NP$-hardness of membership testing in the minrank variety, hence we establish the $\NP$-hardness of the orbit closure containment problem for 3-tensors. Algebraic natural proofs were recently introduced by Forbes, Shpilka and Volk and independently by Grochow, Kumar, Saks and Saraf. Bläser et al. gave a version of an algebraic natural proof barrier for the matrix completion problem which relies on $\coNP \subseteq \exists \BPP$. It implied that constructing equations for the corresponding variety should be hard. We generalize their approach to work with any family of varieties for which the membership problem is $\NP$-hard and for which we can efficiently generate a dense subset. Therefore, a similar barrier holds for the slice rank and the minrank varieties, too. This allows us to set up the slice rank and the minrank varieties as a test-bed for geometric complexity theory (GCT). We determine the stabilizers of the tensors that generate the orbit closures of the two varieties and prove that these tensors are almost characterized by their symmetries. We prove several nontrivial equations for both the varieties using different GCT methods. Many equations also work in the regime where membership testing in the slice rank or minrank varieties is $\NP$-hard. We view this as a promising sign that the GCT approach might indeed be successful.
In this paper, we compute an explicit matrix factorization of a rank 9 Ulrich sheaf on a general cubic hypersurface of dimension at most 7, whose existence was proved by Manivel. Instead of using invariant theory, we use Shamash's construction with a cone over the spinor variety. We also describe an algebro-geometric interpretation of our matrix factorization which connects the spinor tenfold and the Cartan cubic.
We study the Tate resolutions and the maximal Cohen-Macaulay approximations of Cohen-Macaulay modules over Gorenstein rings. One consequence is an extension of a well-known result about linkage of complete intersections.
We describe the Macaulay2 package TateOnProducts and its capabilities, which include computing cohomology tables and Beilinson monads of sheaves on products of projective spaces and the derived category pushForward of a sheaf under a morphism from a projective scheme to a projective space.
Let $S$ be a regular local ring with residue field $k$ and let $M$ be a finitely generated $S$-module. Suppose that $f_1,\dots ,f_c\in S$ is a regular sequence that annihilates $M$, and let $E$ be an exterior algebra over $k$ generated by $c$ elements. The homotopies for the $f_{i}$ on a free resolution of $M$ induce a natural structure of graded $E$-module on ${\rm Tor}^{S}(M,k)$. In the case where $M$ is a high syzygy over the complete intersectionR:=S/(f_{1},\dots,f_{c})$ we describe this $E$-module structure in detail, including its minimal free resolution over $E$. Turning to ${\rm Ext}_{R}(M,\, k)$ we show that, when $M$ is a high syzygy over $R$, the minimal free resolution of ${\rm Ext}_{R}(M,\, k)$ as a module over the ring of CI operators is the Bernstein-Gel'fand-Gel'fand dual of the $E$-module ${\rm Tor}^{S}(M,\,k)$. For the proof we introduce \emph{higher CI operators}, and give a construction of a (generally non-minimal) resolution of $M$ over $S$ starting from a resolution of $M$ over $R$ and its higher CI operators.
Let $C$ be a curve in $\mathbb{P}^4$ and $X$ be a hypersurface containing it. We show how it is possible to construct a matrix factorization on $X$ from the pair $(C,X)$ and, conversely, how a matrix factorization on $X$ leads to curves lying on $X$. We use this correspondence to prove the unirationality of the Hurwitz space $\mathcal{H}_{12,8}$ and the uniruledness of the Brill-Noether space $\mathcal{W}^1_{13,9}$. Several unirational families of curves of genus $16 \leq g \leq 20$ in $\mathbb{P}^4$ are also exhibited.
Singular value decompositions of matrices are widely used in numerical linear algebra with many applications. In this paper, we extend the notion of singular value decompositions to finite complexes of real vector spaces. We provide two methods to compute them and present several applications.
We describe the equations and Gröbner bases of some degenerate K3 surfaces associated to rational normal scrolls. These K3 surfaces are members of a class of interesting singular projective varieties we call correspondence scrolls. The ideals of these surfaces are nested in a simple way that allows us to analyze them inductively. We describe explicit Gröbner bases and syzygies for these objects over the integers and this lets us treat them in all characteristics simultaneously.