ACM Bundles Research Articles

ACM bundles on a general K3 surface of degree 2

We shall study semi-stable aCM bundles on a K3 surface of degree 2 and Picard number 1.

Elliptic curves, ACM bundles and Ulrich bundles on prime Fano threefolds

Let X be any smooth prime Fano threefold of degree 2g-2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2g-2$$\\end{document} in Pg+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb P}^{g+1}$$\\end{document}, with g∈{3,…,10,12}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g \\in \\{3,\\ldots ,10,12\\}$$\\end{document}. We prove that for any integer d satisfying g+32⩽d⩽g+3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left\\lfloor \\frac{g+3}{2} \\right\\rfloor \\leqslant d \\leqslant g+3$$\\end{document} the Hilbert scheme parametrizing smooth irreducible elliptic curves of degree d in X is nonempty and has a component of dimension d, which is furthermore reduced except for the case when (g,d)=(4,3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(g,d)=(4,3)$$\\end{document} and X is contained in a singular quadric. Consequently, we deduce that the moduli space of rank–two slope–stable ACM bundles Fd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal F}_d$$\\end{document} on X such that det(Fd)=OX(1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\det ({\\mathcal F}_d)={\\mathcal O}_X(1)$$\\end{document}, c2(Fd)·OX(1)=d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c_2({\\mathcal F}_d)\\cdot {\\mathcal O}_X(1)=d$$\\end{document} and h0(Fd(-1))=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h^0({\\mathcal F}_d(-1))=0$$\\end{document} is nonempty and has a component of dimension 2d-g-2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2d-g-2$$\\end{document}, which is furthermore reduced except for the case when (g,d)=(4,3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(g,d)=(4,3)$$\\end{document} and X is contained in a singular quadric. This completes the classification of rank–two ACM bundles on prime Fano threefolds. Secondly, we prove that for every h∈Z+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h \\in {\\mathbb Z}^+$$\\end{document} the moduli space of stable Ulrich bundles E\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal E}$$\\end{document} of rank 2h and determinant OX(3h)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal O}_X(3h)$$\\end{document} on X is nonempty and has a reduced component of dimension h2(g+3)+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h^2(g+3)+1$$\\end{document}; this result is optimal in the sense that there are no other Ulrich bundles occurring on X. This in particular shows that any prime Fano threefold is Ulrich wild.

Hodge rank of ACM bundles and Franchetta's conjecture

Hodge rank of ACM bundles and Franchetta's conjecture

Derived Category and ACM Bundles of Moduli Space of Vector Bundles on a Curve

Abstract We show that the derived category of a curve is embedded into the derived category of the moduli space of vector bundles on the curve of coprime rank and degree. We also generalize the semiorthogonal decomposition constructed by Narasimhan and Belmans-Mukhopadhyay. Finally, we produce a one-dimensional family of ACM bundles over the moduli space.

Ulrich bundles on cubic fourfolds

We show the existence of rank 6 Ulrich bundles on a smooth cubic fourfold. First, we construct a simple sheaf \mathcal E of rank 6 as an elementary modification of an ACM bundle of rank 6 on a smooth cubic fourfold. Such an \mathcal E appears as an extension of two Lehn–Lehn–Sorger–van Straten sheaves. Then we prove that a general deformation of \mathcal E(1) becomes Ulrich. In particular, this says that general cubic fourfolds have Ulrich complexity 6.

On the base case of a conjecture on ACM bundles over hypersurfaces

On the base case of a conjecture on ACM bundles over hypersurfaces

On initialized and ACM line bundles over a smooth sextic surface in

Let be a smooth hypersurface of degree 6 over complex numbers. In this article, we give a characterization of initialized and ACM bundles of rank 1 on X with respect to the line bundle given by a smooth hyperplane section of X.

Rank two ACM bundles on the double embedding of a quadric surface

In this article, we completely classify locally free sheaves of rank 2 such that for on the double embedding of a smooth quadric surface and prove the existence of them. Geometric properties of the sets of points corresponding to by Serre correspondence are obtained.

ACM bundles of rank 2 on quartic hypersurfaces in $${\mathbb {P}}^3$$ and Lazarsfeld-Mukai bundles

Let X be a smooth quartic hypersurface in $$\mathbb {P}^3$$ . By the Brill-Noether theory of curves on K3 surfaces, if a rank 2 aCM bundle on X is globally generated, then it is the Lazarsfeld-Mukai bundle $$E_{C,Z}$$ associated with a smooth curve C on X and a base point free pencil Z on C. In this paper, we will focus on the classification of such bundles on X to investigate aCM bundles of rank 2 on X. Concretely, we will give a necessary condition for a rank 2 vector bundle of type $$E_{C,Z}$$ to be indecomposable initialized and aCM, in the case where the class of C in $${{\,\mathrm{Pic}\,}}(X)$$ is contained in the sublattice of rank 2 generated by the hyperplane class of X and a non-trivial initialized aCM line bundle on X.

Non-Ulrich representation type

We show that a smooth projective non-degenerate arithmetically Cohen-Macaulay subvariety X of P^N infinite Cohen-Macaulay type becomes of finite Cohen-Macaulay type by removing Ulrich bundles if and only if N = 5 and X is a quartic scroll or the Segre product of a line and a plane. In turn, we give a complete and explicit classification of ACM bundles over these varieties.

ACM bundles on a general abelian surface

In this note, we shall study semistable aCM bundles on an abelian surface of Picard number 1.

ACM line bundles on polarized K3 surfaces

An ACM bundle on a polarized algebraic variety is defined as a vector bundle whose intermediate cohomology vanishes. We are interested in ACM bundles of rank one with respect to a very ample line bundle on a K3 surface. In this paper, we give a necessary and sufficient condition for a non-trivial line bundle $\mathcal{O}_X(D)$ on $X$ with $|D|=\emptyset$ and $D^2\geq L^2-6$ to be an ACM and initialized line bundle with respect to $L$, for a given K3 surface $X$ and a very ample line bundle $L$ on $X$.

A remark on Ulrich and ACM bundles

I show that on any smooth, projective ordinary curve of genus at least two and a projective embedding, there is a natural example of a stable Ulrich bundle for this embedding: namely the sheaf BX1 of locally exact differentials twisted by OX(1) given by this embedding and in particular there exist ordinary varieties of any dimension which carry Ulrich bundles. In higher dimensions, assuming X is Frobenius split variety I show that BX1 is an ACM bundle and if X is also a Calabi–Yau variety and p>2 then BX1 is not a direct sum of line bundles. In particular I show that BX1 is an ACM bundle on any ordinary Calabi–Yau variety. I also prove a characterization of projective varieties with trivial canonical bundle such that BX1 is ACM (for some projective embedding datum): all such varieties are Frobenius split (with trivial canonical bundle).

Ulrich and aCM bundles from invariant theory

We use certain special prehomogeneous representations of algebraic groups in order to construct aCM vector bundles, possibly Ulrich, on certain families of hypersurfaces. Among other results, we show that a general cubic hypersurface of dimension seven admits an indecomposable Ulrich bundle of rank nine, and that a general quartic fourfold admits an unsplit aCM bundle of rank six.

Remarks on higher-rank ACM bundles on hypersurfaces

In terms of the number of generators, one of the simplest non-split rank-3 arithmetically Cohen–Macaulay bundles on a smooth hypersurface in P5 is 6-generated. We prove that a general hypersurface in P5 of degree d≥3 does not support such a bundle. We also prove that a smooth positive dimensional hypersurface in projective space of even degree does not support an Ulrich bundle of odd rank and determinant of the form OX(c) for some integer c. This verifies some cases of conjectures we discuss here.

Generalized twisted cubics on a cubic fourfold as a moduli space of stable objects

We revisit the work of Lehn–Lehn–Sorger–van Straten on twisted cubic curves in a cubic fourfold not containing a plane in terms of moduli spaces. We show that the blow-up Z′ along the cubic of the irreducible holomorphic symplectic eightfold Z, described by the four authors, is isomorphic to an irreducible component of a moduli space of Gieseker stable torsion sheaves or rank three torsion free sheaves.For a very general such cubic fourfold, we show that Z is isomorphic to a connected component of a moduli space of tilt-stable objects in the derived category and to a moduli space of Bridgeland stable objects in the Kuznetsov component. Moreover, the contraction between Z′ and Z is realized as a wall-crossing in tilt-stability.Finally, Z is birational to an irreducible component of a moduli space of Gieseker stable aCM bundles of rank six.

Rank two aCM bundles on the del Pezzo fourfold of degree 6 and its general hyperplane section

In the present paper we completely classify locally free sheaves of rank 2 with vanishing intermediate cohomology modules on the image of the Segre embedding P2×P2⊆P8 and its general hyperplane sections.Such a classification extends similar already known results regarding del Pezzo varieties with Picard numbers 1 and 3 and dimension at least 3.

Surfaces of minimal degree of tame representation type and mutations of Cohen–Macaulay modules

We provide two examples of smooth projective surfaces of tame CM type, by showing that the parameter space of isomorphism classes of indecomposable ACM bundles with fixed rank and determinant on a rational quartic scroll in P5 is either a single point or a projective line. These turn out to be the only smooth projective ACM varieties of tame CM type besides elliptic curves, [1].For surfaces of minimal degree and wild CM type, we classify rigid Ulrich bundles as Fibonacci extensions. For F0 and F1, embedded as quintic or sextic scrolls, a complete classification of rigid ACM bundles is given.

Rank two aCM bundles on the del Pezzo threefold of degree 7

We classify indecomposable aCM bundles of rank 2 on the del Pezzo threefold of degree 7 and analyze their corresponding moduli spaces.

On rank two aCM bundles

ABSTRACTLet (F,𝒪F(1)) be a smooth polarized projective variety of dimension n. In the present paper, we prove some easy results on aCM bundles of rank 2 on F, i.e., locally free sheaves ℰ of rank 2 on F such that hi(F,ℰ(t)) = 0, for i = 1,…,n−1 and t∈ℤ.We obtain some results in the case of a very ample polarization when n≥5. We apply these results in order to deal with bundles on non-degenerate complete intersection with degree up to 9.A complete description is given for the complete intersections of two quadrics when n≥4.