Scott Nollet

The Hilbert schemes of degree three curves

In this paper we show that the Hilbert scheme H(3,g) of locally Cohen-Macaulay curves in ℙ3 of degree three and genus g is connected. This is achieved by giving a classification of these curves, determining the irreducible components of H(3,g), and giving certain specializations to show connectedness. As a byproduct, we find that there are curves which lie in the closure of each irreducible component.

Integral subschemes of codimension two

Abstract An even linkage class L of two-codimensional subschemes in P n has a natural partial ordering given by domination. In this paper we give a necessary condition for X ∈ L to be integral in terms of its location in the poset structure on L . The condition is almost sufficient in the sense that if a subscheme dominates an integral subscheme and satisfies the necessary conditions, then it can be deformed with constant cohomology to an integral subscheme. In particular, the necessary conditions are sufficient in the case that Lazarsfeld and Rao originally studied, since the minimal element for L was a smooth connected space curve.

Noether–Lefschetz Theorem with Base Locus

For an arbitrary curve (possibly reducible, non-reduced, of mixed dimension) lying on a normal surface, the general surface S of high degree containing Z is also normal but often singular. We compute the class groups of the very general such surface, thereby extending the Noether–Lefschetz theorem (the special case when Z is empty). Our method is an adaptation of Griffiths and Harris’ degeneration proof, simplified by a cohomology and base change argument. We give applications to computing Picard groups. Dedicated to Robin Hartshorne on his 70th birthday

Birationality of étale maps via surgery

Abstract We use a counting argument and surgery theory to show that if D is a sufficiently general algebraic hypersurface in , then any local diffeomorphism F : X → of simply connected manifolds which is a d-sheeted cover away from D has degree d = 1 or d = ∞ (however all degrees d > 1 are possible if F fails to be a local diffeomorphism at even a single point). In particular, any étale morphism F : X → of algebraic varieties which covers away from such a hypersurface D must be birational.

Degrees of generators of ideals defining curves in projective space

For an arithmetically Cohen–Macaulay subscheme of projective space, there is a well-known bound for the highest degree of a minimal generator for the defining ideal of the subscheme, in terms of the Hilbert function. We prove a natural extension of this bound for arbitrary locally Cohen–Macaulay subschemes. We then specialize to curves in P 3, and show that the curves whose defining ideals have generators of maximal degree satisfy an interesting cobomological property. The even liaison classes possessing such curves are characterized, and we show that within an even liaison class, all curves with the property satisfy a strong Lazarsfeld–Rao structure theorem. This allows us to give relatively complete conditions for when a liaison class contains curves whose ideals have maximal degree generators, and where within the liaison class they occur.

Detaching embedded points

Suppose that XYP N differ at finitely many points: when is Y a flat limit of X union isolated points? Our main result says that this is possible when X is a local complete intersection of codimension 2 and the multiplicities are at most 3. We show by example that no hypothesis can be weakened: the conclusion fails for (a) embedded points of multiplicity > 3, (b) local complete intersections X of codimension > 2 and (c) non-local complete intersections of codimension 2. As applications, we determine the irreducible components of Hilbert schemes of space curves with high arithmetic genus and show the smoothness of the Hilbert component whose general member is a plane curve union a point in P 3 .

Hilbert Scheme of a Pair of Codimension Two Linear Subspaces

We study the component H n of the Hilbert scheme whose general point parameterizes a pair of codimension two linear subspaces in ℙ n for n ≥ 3. We show that H n is smooth and isomorphic to the blow-up of the symmetric square of 𝔾(n − 2, n) along the diagonal. Further H n intersects only one other component in the full Hilbert scheme, transversely. We determine the stable base locus decomposition of its effective cone and give modular interpretations of the corresponding models, hence conclude that H n is a Mori dream space.

Bounds on the Rao function

Abstract In this article we find upper bounds on the Rao function for space curves in terms of the degree, genus and the minimal degree s of a surface which contains the curve. These bounds are shown to be sharp for s≤4. This paper is dedicated to David Buchsbaum on the occasion of his 70th birthday.

Smooth curves linked to thick curves

In this paper we construct smooth irreducible spacecurves C which link geometrically by surfaces of minimal degree containingC to curves Γ of generic embedding dimension three. This produces interesting behavior with respect to both C and Γ. The curvesΓ link to smooth connected curves by surfaces of low degree butcannot link to smooth connected curves by surfaces of high degree.The curves C give counterexamples to a conjecture of Martin-Deschamps and Perrin.

On curve sections of rank two reflexive sheaves

Let F be a normalized rank 2 reflexive sheaf on P3 with Chern classes c 1,c 2,c 3. Let α be the least integer such that 0≠H 0 F(α) and β be the smallest integer such that H 0 F(n) has sections whose zero scheme is a curve for all n≥ β . We show that if T 0 is the largest root of the cubic polynomial then β ≥ T 0-α-c 1-1. There are applications to the smallest degree of a surface containing a curves which are the zero schemes of sections of H 0 F(α).