119313872
Moduli spaces of rank two aCM bundles on the Segre product of three projective lines
Let P^n be the projective space of dimension n on an algebraically closed field of characteristic 0 and F be the image of the Segre embedding of P^1xP^1xP^1 inside P^7. In the present paper we deal with the moduli spaces of locally free sheaves E on F of rank 2 with h^i(F,E(t))=0 for i=1,2 and each integer t.
121404553
Spinor bundles on quadrics
We define some stable vector bundles on the complex quadric hypersurface Qn of dimension n as the natural generalization of the universal bundle and the dual of the quotient bundle on Q4 ~ Gr(l,3). We call them spinor bundles. When n = 2fc — 1 there is one spinor bundle of rank 2k~1. When n = 2k there are two spinor bundles of rank 2k~1. Their behavior is slightly different according as n = 0 (mod 4) or n = 2 (mod 4). As an application, we describe some moduli spaces of rank 3 vector bundles on Q5 and Qe- Introduction. Let Qn be the smooth quadric hypersurface of the complex pro- jective space Pn+1. In this paper we define in a geometrical way some vector bundles on the quadric Qn as the natural generalization of the universal bundle and the dual of the quotient bundle on Q4 ~ Gr(l,3). We call them spinor bundles. On Q4 this definition is equivalent to the usual one. Spinor bundles are homogeneous and stable (according to the definition of Mum- ford-Takemoto). We study their first properties using the geometrical description given and some standard techniques available in (OSS). We also use a theorem of Ramanan (see (Um)) about the stability of homoge- neous bundles induced by irreducible representations. When n is odd there is only one spinor bundle, while when n is even there are two nonisomorphic spinor bun- dles. When n is even the behavior of spinor bundles is slightly different according as n = 0 (mod4) or n = 2 (mod4). In (Ot2) we have given a cohomological splitting criterion for vector bundles on quadrics involving spinor bundles. Qn ~ Spin(n + 2)/P(cty) (St) is a homogeneous manifold, and the semisimple part of the Lie algebra of -P(ai) is o(n). At the level of Lie algebras, spinor bundles are defined from the spin and half-spin representations of o(n). The paper is divided into three sections. In §1 we give some preliminary results and we define the spinor bundles. In §2 we study the first properties of spinor bundles. In §3, as an application, we describe some moduli spaces of rank 3 vector bundles on Q5 and Qq.
8955060
The classification of homogeneous Cohen-Macaulay rings of finite representation type
In this paper we classify the homogeneous Cohen-Macaulay rings which are of finite representation type, that is, the Cohen-Macaulay rings which are positively graded and generated in degree 1, with an algebraically closed field of characteristic 0 in degree 0, and which have, up to isomorphisms and shifts in the grading, only a finite number of indecomposable maximal Cohen-Macaulay modules (MCM-modules). Our main contribution is to show that a homogeneous Cohen-Macaulay ring of finite representation type and dimension >__ 2 must have "minimal multiplicity". Putting this together with previous results of Auslander, Auslander-Reiten, Buchweitz-Greuel-Schreyer, Greuel-Kn6rrer, and Solberg, we obtain:
5505177
THE GEOMETRY OF ULRICH BUNDLES ON DEL PEZZO SURFACES
Abstract Given a smooth del Pezzo surface X d ⊆ P d of degree d, we isolate the essential geometric obstruction to a vector bundle on X d being an Ulrich bundle by showing that an irreducible curve D of degree dr on X d represents the first Chern class of a rank-r Ulrich bundle on X d if and only if the kernel bundle of the general smooth element of | D | admits a generalized theta-divisor. Moreover, we show that any smooth arithmetically Gorenstein surface whose Ulrich bundles admit such a characterization is necessarily del Pezzo. This result is applied to produce new examples of complete intersection curves with semistable kernel bundle, and also combined with work of Farkas, Mustaţǎ and Popa to relate the existence of Ulrich bundles on X d to the Minimal Resolution Conjecture for curves lying on X d . In particular, we show that a smooth irreducible curve D of degree 3r lying on a smooth cubic surface X 3 represents the first Chern class of an Ulrich bundle on X 3 if and only if the Minimal Resolution Conjecture holds for the general smooth element of | D | .
15534703
Rank 2 arithmetically Cohen–Macaulay bundles on a nonsingular cubic surface
Abstract Rank 2 indecomposable arithmetically Cohen–Macaulay bundles E on a nonsingular cubic surface X in P 3 are classified, by means of the possible forms taken by the minimal graded free resolution of E over P 3 . The admissible values of the Chern classes of E are listed and the vanishing locus of a general section of E is studied. Properties of E such as slope (semi)stability and simplicity are investigated; the number of relevant families is computed together with their dimension.
51763327
Varieties of small codimension in projective space
Introduction. I would like to begin by stating a conjecture. While I am not convinced of the truth of this statement, I think it is useful to crystallize one's ideas, and to have a particular problem in mind. Then for the remainder of the talk, I propose to examine this question in a rather general way from a number of different perspectives. This will give me an opportunity to report on recent work in several areas of algebraic geometry, and at the same time to mention a number of open problems. Let k be an algebraically closed field. Let P be the «-dimensional projective space over k. Let 7 g P n b e a nonsingular subvariety of dimension r. We say that F is a complete intersection in P if one can find n—r hypersurfaces Hl9 • • • , Hn_r, such that Y=H1ri• -C\Hn_r, and such that this intersection is transversal, i.e. the hypersurfaces Hi are nonsingular at all points of F, and their tangent hyperplanes intersect properly at each point of F In algebraic terms, F is a complete intersection if and only if its homogeneous prime ideal I(Y)^k[x0, • • • , xn] can be generated by exactly n—r homogeneous polynomials. Conjecture. If Y is a nonsingular subvariety of dimension r of P n , and if r>§« , then F is a complete intersection. The paper is divided into six sections : §1. Representing cohomology classes by subvarieties. §2. Cohomological properties of the subvariety. §3. Examples. Subvarieties of small degree. §4. Embedding varieties in projective space. §5. Connections with local algebra. §6. Existence of vector bundles on P.
27226449
A home-made Hartshorne-Serre correspondence
We provide an elementary proof of the Hartshorne-Serre correspondence for constructing vector bundles from local complete intersection subschemes of codimension two. This will be done, as in the correspondence of hypersurfaces and line bundles, by patching together local determinantal equations in order to produce sections of a vector bundle.
209835137
The geometry of moduli spaces of sheaves
Preface to the second edition Preface to the first edition Introduction Part I. General Theory: 1. Preliminaries 2. Families of sheaves 3. The Grauert-Mullich Theorem 4. Moduli spaces Part II. Sheaves on Surfaces: 5. Construction methods 6. Moduli spaces on K3 surfaces 7. Restriction of sheaves to curves 8. Line bundles on the moduli space 9. Irreducibility and smoothness 10. Symplectic structures 11. Birational properties Glossary of notations References Index.
119279522
Instanton bundles on Fano threefolds
We introduce the notion of an instanton bundle on a Fano threefold of index 2. For such bundles we give an analogue of a monadic description and discuss the curve of jumping lines. The cases of threefolds of degree 5 and 4 are considered in a greater detail.
197491170
Even and odd instanton bundles on Fano threefolds of Picard number 1
We consider an analogue of the notion of instanton bundle on the projective 3-space, consisting of a class of rank-2 vector bundles defined on smooth Fano threefolds X of Picard number one, having even or odd determinant according to the parity of K_X. 
We first construct a well-behaved irreducible component of their moduli spaces. Then, when the intermediate Jacobian of X is trivial, we look at the associated monads, hyperdeterminants and nets of quadrics. 
We also study one case where the intermediate Jacobian of X is non-trivial, namely when X is the intersection of two quadrics in P5, relating instanton bundles on X to vector bundles of higher rank on a the curve of genus 2 naturally associated with X.
119315625
Rank two aCM bundles on the del Pezzo threefold with Picard number 3
Let k be an algebraically closed field of characteristic 0. A del Pezzo threefold F with maximal Picard number is isomorphic to P^1xP^1xP^1, where P^1 is the projective line over k. In the present paper we completely classify locally free sheaves of rank 2 with vanishing intermediate cohomology over such an F. Such a classification extends similar results proved by E. Arrondo and L. Costa regarding del Pezzo threefolds with Picard number 1.
119177707
STABLE ULRICH BUNDLES
The existence of stable ACM vector bundles of high rank on algebraic varieties is a challenging problem. In this paper, we study stable Ulrich bundles (that is, stable ACM bundles whose corresponding module has the maximum number of generators) on nonsingular cubic surfaces X ⊂ ℙ3. We give necessary and sufficient conditions on the first Chern class D for the existence of stable Ulrich bundles on X of rank r and c1 = D. When such bundles exist, we prove that the corresponding moduli space of stable bundles is smooth and irreducible of dimension D2 - 2r2 + 1 and consists entirely of stable Ulrich bundles (see Theorem 1.1). We are also able to prove the existence of stable Ulrich bundles of any rank on nonsingular cubic threefolds in ℙ4, and we show that the restriction map from bundles on the threefold to bundles on the surface is generically etale and dominant.
8125077
Resultants and Chow forms via exterior syzygies
Let W be a vector space of dimension n+ 1 over a field K. The Chow divisor of a k-dimensional variety X in P = P(W ) is the hypersurface, in the Grassmannian Gk+1 of planes of codimension k+1 in P, whose points (over the algebraic closure of K) are the planes that meet X . The Chow form of X is the defining equation of the Chow divisor. For example, the resultant of k+ 1 forms of degree e in k + 1 variables is the Chow form of P embedded by the e-th Veronese mapping in P with n = ( k+e k ) − 1. More generally, the Chow divisor of a k-cycle ∑ i ni[Vi] on projective space is defined to be ∑ i niDi, where Di is the Chow divisor of Vi. The Chow divisor of a sheaf F with k-dimensional support is the Chow divisor of the associated k-cycle of F . In this paper we will give a new expression for the Chow divisor and apply it to give explicit formulas in many new cases. Starting with a sheaf F on P, we use exterior algebra methods to define a canonical and effectively computable Chow complex of F on each Grassmannian of planes in P. If F has k-dimensional support, we show that the Chow form of F is the determinant of the Chow complex of F on the Grassmannian of planes of codimension k + 1. The Beilinson monad of F [Beilinson 1978] is the Chow complex of F on the Grassmannian of 0-planes (that is, on P itself.) In particular, we are able to give explicit determinantal and Pfaffian formulas for resultants in some cases where no polynomial formulas were known. For example, the Horrocks-Mumford bundle gives rise to polynomial formulas for the resultant of five homogeneous forms of degrees 4, 6 or 8 in five variables. The easiest of our new formulas to write down is for the resultant of 3 quadratic forms in three variables, the Chow form of the Veronese surface in P. Using the tangent bundle of P, conclude that it can be written in “Bezout form” (described below) as the Pfaffian of the matrix  0 [245] [345] [135] [045] [035] [145] [235]
120064462
On the genus and Hartshorne-Rao module of projective curves
Abstract. In this paper optimal upper bounds for the genus and the dimension of the graded components of the Hartshorne-Rao module of curves in projective n-space are established. This generalizes earlier work by Hartshorne [H] and Martin-Deschamps and Perrin [MDP]. Special emphasis is put on curves in 
${\bf P}^4$. The first main result is a so-called Restriction Theorem. It says that a non-degenerate curve of degree 
$d \geq 4$ in 
${\bf P}^4$ over a field of characteristic zero has a non-degenerate general hyperplane section if and only if it does not contain a planar curve of degree 
$d-1$ (see Th. 1.3). Then, using methods of Brodmann and Nagel, bounds for the genus and Hartshorne-Rao module of curves in 
${\bf P}^n$ with non-degenerate general hyperplane section are derived. It is shown that these bounds are best possible in a very strict sense. Coupling these bounds with the Restriction Theorem gives the second main result for curves in 
${\bf P}^4$. Then curves of maximal genus are investigated. The Betti numbers of their minimal free resolutions are computed and a description of all reduced curves of maximal genus in 
${\bf P}^n$ of degree 
$\geq n+2$ is given. Finally, all pairs (d,g) of integers which really occur as the degree d and genus g of a non-degenerate curve in 
${\bf P}^4$ are described.
197660097
Algebraic geometry
The name Algebraic Geometry comes from the fact that in this part of Mathematics one tries to study geometric objects (mainly) through algebraic techniques. This combination of algebra and geometry is extremely fruitful, and as a result the field of Algebraic Geometry has become big and very diverse. There are many connections to other areas/techniques in mathematics, such as Number Theory, Differential Geometry, Topology, Category Theory, Cryptography, Mathematical Physics, and so on. All in all, it’s better to think of Algebraic Geometry as indicating a sub-area of mathematics as a whole, rather than a very precisely defined subfield.
55819018
Compact moduli spaces for slope-semistable sheaves
We resolve pathological wall-crossing phenomena for moduli spaces of sheaves on higher-dimensional base manifolds. This is achieved by considering slope-semistability with respect to movable curves rather than divisors. Moreover, given a projective n-fold and a curve C that arises as the complete intersection of n-1 very ample divisors, we construct a modular compactification of the moduli space of vector bundles that are slope-stable with respect to C. Our construction generalises the algebro-geometric construction of the Donaldson-Uhlenbeck compactification by Joseph Le Potier and Jun Li. Furthermore, we describe the geometry of the newly construced moduli spaces by relating them to moduli spaces of simple sheaves and to Gieseker-Maruyama moduli spaces.