Robin Hartshorne

Ample vector bundles

© Publications mathématiques de l’I.H.É.S., 1966, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

On the de rham cohomology of algebraic varieties

© Publications mathématiques de l’I.H.É.S., 1966, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Connectedness of the Hilbert scheme

© Publications mathématiques de l’I.H.É.S., 1966, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

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 ones 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.

Ample vector bundles on curves

In our earlier paper [4] we developed the basic sheaftheoretic and cohomological properties of ample vector bundles. These generalize the corresponding well-known results for ample line bundles. The numerical properties of ample vector bundles are still poorly understood. For line bundles, Nakai’s criterion characterizes ampleness by the positivity of certain intersection numbers of the associated divisor with subvarieties of the ambient variety. For vector bundles, one would like to characterize ampleness by the numerical positivity of the Chern classes of the bundle (and perhaps of its restrictions to subvarieties and their quotients). Such a result, like the Riemann-Roch theorem, giving an equivalence between cohomological and numerical properties of a vector bundle, may be quite subtle. Some progress has been made by Gieseker [2], by Kleiman [8], and in the analytic case, by Griffiths [3].

ACM bundles on cubic surfaces

In this paper we prove that, for every r = 2, the moduli space MsX (r; c1,c2) of rank r stable vector bundles with Chern classes c1 = rH and c2 = 1/2 (3r2 - r) on a nonsingular cubic surface X ? P3 contains a nonempty smooth open subset formed by ACM bundles, i.e. vector bundles with no intermediate cohomology. The bundles we consider for this study are extremal for the number of generators of the corresponding module (these are known as Ulrich bundles), so we also prove the existence of indecomposable Ulrich bundles of arbitrarily high rank on X.

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.

Stable vector bundles and instantons

Methods of abstract algebraic geometry are used to study rank 2 stable vector bundles on ℙ3. These bundles are then used to give self-dual solutions, called instantons, of the Yang-Mills equation onS4.

The genus of space curves

SuntoSi dimostra che seC⊃ℙk3 (k di caratteristica qualunque) é una curva, non piana, localmente Cohen-Macaulay, di gradod, allorapa (C)≤((d−2)(d−3))/2. Si mostra che questa limitazione è ottimale, e si classificano le curve di genere aritmetico massimale.AbstractLetC be a curve contained in ℙk3 (k of any characteristic), which is locally Cohen-Macaulay, not contained in a plane and of degreed. We prove thatpa(C)≤≤((d−2)(d−3))/2. Moreover we show existence of curves with anyd, pa satisfying this inequality and we characterize those curves for which equality holds.

Curves with high self-intersection on algebraic surfaces

© Publications mathématiques de l’I.H.É.S., 1969, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.