arXiv subjects
Publications and source records attributed to Frank-Olaf Schreyer.
Based on computeralgebra experiments we formulate a refined version of Green's conjecture and a conjecture of Schicho-Schreyer-Weimann which conjecturally also holds in positive characteristic. The experiments are done by using our Macaulay2 package, which constructs random canonically embedded curves of genus $g\leq 15$ over arbitrary small finite fields.
In this paper we prove that the Hurwitz space H_{9,8}, which parameterizes 8-sheeted covers of P^1 by curves of genus 9, is unirational. Our construction leads to an explicit Macaulay2 code, which will randomly produce a nodal curve of degree 8 of geometric genus 9 with 12 double points and together with a pencil of degree 8.
We prove an analogue of Horrocks' splitting theorem for Segre-Veronese varieties building upon the theory of Tate resolutions on products of projective spaces.
Based on Schreyer's algorithm (Schreyer, 1980, 1991; Berkesch and Schreyer, 2014), we present two refined algorithms for the computation of syzygies. The two main ideas of the first algorithm, called LiftHybrid, are the following: First, we may leave out certain terms of module elements during the computation which do not contribute to the result. These terms are called "lower order terms", see Definition 4.2. Second, we do not need to order the remaining terms of these module elements during the computation. This significantly reduces the number of monomial comparisons for the arithmetic operations. For the second algorithm, called LiftTree, we additionally cache some partial results and reuse them at the remaining steps.
We describe the Tate resolution of a coherent sheaf or complex of coherent sheaves on a product of projective spaces. Such a resolution makes explicit all the cohomology of all twists of the sheaf, including, for example, the multigraded module of twisted global sections, and also the Beilinson monads of all twists. Although the Tate resolution is highly infinite, any finite number of components can be computed efficiently, starting either from a Beilinson monad or from a multigraded module.
We apply the theory of Groebner bases to the computation of free resolutions over a polynomial ring, the defining equations of a canonically embedded curve, and the unirationality of the moduli space of curves of a fixed genus.
A recent result of Eisenbud-Schreyer and Boij-Söderberg proves that the Betti diagram of any graded module decomposes as a positive rational linear combination of pure diagrams. When does this numerical decomposition correspond to an actual filtration of the minimal free resolution? Our main result gives a sufficient condition for this to happen. We apply it to show the non-existence of free resolutions with some plausible-looking Betti diagrams and to study the semigroup of quiver representations of the simplest "wild" quiver.
We formulate, and in some cases prove, three statements concerning the purity or, more generally the naturality of the resolution of various rings one can attach to a generic curve of genus g and a torsion point of order l in its Jacobian. These statements can be viewed an analogues of Green's Conjecture and we verify them computationally for bounded genus. We then compute the cohomology class of the corresponding non-vanishing locus in the moduli space R_{g,l} of twisted level l curves of genus g and use this to derive results about the birational geometry of R_{g, l}. For instance, we prove that R_{g,3} is a variety of general type when g>11 and the Kodaira dimension of R_{11,3} is greater than or equal to 19. In the last section we explain probabilistically the unexpected failure of the Prym-Green conjecture in genus 8 and level 2.
In this note, we explain how certain matrix factorizations on cubic threefolds lead to families of curves of genus 15 and degree 16 in P^4. We prove that the moduli space M={(C,L) | C a curve of genus 15, L a line bundle on C of degree 16 with 5 sections} is birational to a certain space of matrix factorizations of cubics, and that M is uniruled. Our attempt to prove the unirationality of this space failed with our methods. Instead one can interpret our findings as evidence for the conjecture that the basis of the maximal rational connected fibration of M has a three dimensional base.
We develop in this article an algorithm that, given a projective curve $C$, computes a \textit{gonal map}, that is, a finite morphism from $C$ to the projective line of minimal degree. Our method is based on the computation of scrollar syzygies of canonical curves. We develop an improved version of our algorithm for curves with a unique gonal map and we discuss a characterization of such curves in terms of Betti numbers. Finally, we derive an efficient algorithm for radical parametrization of curves of gonality $\le 4$.
Using the possibility of computationally determining points on a finite cover of a unirational variety over a finite field, we determine all possibilities for direct Gorenstein linkages between general sets of points in P^3 over an algebraically closed field of characteristic 0. As a consequence we show that a general set of d points is glicci (that is, in the Gorenstein linkage class of a complete intersection) if d <= 33 or d=37,38. Computer algebra plays an essential role in the proof. The case of 20 points had been an outstanding problem in the area for a dozen years.
The $G$-parking function ideal $M_G$ of a directed multigraph $G$ is a monomial ideal which encodes some of the combinatorial information of $G$. It is an initial ideal of the toppling ideal $I_G$, a lattice ideal intimately related to the chip-firing game on a graph. Both ideals were first studied by Cori, Rossin, and Salvy. A minimal free resolution for $M_G$ was given by Postnikov and Shaprio in the case when $G$ is saturated, i.\,e., whenever there is at least one edge $(u,v)$ for every ordered pair of distinct vertices $u$ and $v$. They also raised the problem of an explicit description of the minimal free resolution in the general case. In this paper, we give a minimal free resolution of $M_G$ for any undirected multigraph $G$, as well as for a family of related ideals including the toppling ideal $I_G$. This settles a conjecture of Manjunath and Sturmfels, as well as a conjecture of Perkinson and Wilmes.
A collection of n distinct hyperplanes L_i ={l_i=0} in P^{n-1}, the n-1-dimensional projective space over an algebraically closed field of characteristic not equal to 2, is a polar simplex of a quadric Q={q=0}, if each L_i is the polar hyperplane of the point p_i, the intersection point of the L_j with j different from i, equivalently, if q= l_1^2+...+l_n^2 for suitable choices of the linear forms l_i. In this paper we study the closure VPS(Q,n) in Hilb_n(P^{n-1}) of the variety of sums of powers presenting Q from a global viewpoint: VPS(Q,n) is a smooth Fano variety of index 2 and Picard number 1 when n<6, and VPS(Q,n) is singular when n>= 6.
We illustrate the use of the computer algebra system Macaulay2 for simplifications of classical unirationality proofs. We explicitly treat the moduli spaces of curves of genus g=10, 12 and 14.
We give sharp bounds on the vanishing of the cohomology of a tensor product of vector bundles on the n-dimensional projective space in terms of the vanishing of the cohomology of the factors. For this purpose we introduce regularity indices generalizing the Castelnuovo-Mumford regularity. As an application we give a sufficient condition for a vector bundle to have an unobstructed deformation theory that depends only on the cohomology table of the bundle. We construct complete families of bundles with such cohomology tables.
We give a lower bound for the degree of a finite apolar subscheme of a symmetric form F, in terms of the degrees of the generators of the annihilator ideal of F. In the special case, when F is a monomial x_0^d_0 x_2^d_2... x_n^d_n with d_0<= d_1<=...<=d_n-1<= d_n we deduce that the minimal length of an apolar subscheme of F is (d_0+1)...(d_n-1+1), and if d_0=..=d_n, then this minimal length coincides with the rank of F.
The variety of sums of powers of a homogeneous polynomial of degree d in n variables is defined and investigated in some examples, old and new. These varieties are studied via apolarity and syzygies. Classical results of Sylvester (1851), Hilbert (1888), Dixon and Stuart (1906) and some more recent results of Mukai (1992) are presented together with new results for the cases (n,d)=(3,8), (4,2), (5,3). In the last case the variety of sums of 8 powers of a general cubic form is a Fano 5-fold of index 1 and degree 660.
The Betti numbers of a graded module over the polynomial ring form a table of numerical invariants that refines the Hilbert polynomial. A sequence of papers sparked by conjectures of Boij and Söderberg have led to the characterization of the possible Betti tables up to rational multiples---that is, to the rational cone generated by the Betti tables. We will summarize this work by describing the cone and the closely related cone of cohomology tables of vector bundles on projective space, and we will give new, simpler proofs of some of the main results. We also explain some of the applications of the theory, including the one that originally motivated the conjectures of Boij and Söderberg, a proof of the Multiplicity Conjecture of Herzog, Huneke and Srinivasan.