arXiv subjects

Scott Nollet

Publications and source records attributed to Scott Nollet.

15 recordsLinked to original sources

arXiv2025

Smoothing surfaces on fourfolds

If $\mathcal E, \mathcal F$ are vector bundles of ranks $r-1,r$ on a smooth fourfold $X$ and $\mathcal{Hom}(\mathcal E,\mathcal F)$ is globally generated, it is well known that the general map $\phi: \mathcal E \to \mathcal F$ is injective and drops rank along a smooth surface. Chang improved on this with a filtered Bertini theorem. We strengthen these results by proving variants in which (a) $\mathcal F$ is not a vector bundle and (b) $\mathcal{Hom}(\mathcal E,\mathcal F)$ is not globally generated. As an application, we give examples of even linkage classes of surfaces on $\mathbb P^4$ in which all integral surfaces are smoothable, including the linkage classes associated with the Horrocks-Mumford surface.

math.AG↗

arXiv2016

Geometric Divisors in Normal Local Domains

Let A be the local ring at a point of a normal complex variety with completion R. Srinivas has asked about the possible images of the induced map from Cl A to Cl R over all geometric normal domains A with fixed completion R. We use Noether-Lefschetz theory to prove that all finitely generated subgroups are possible in some familiar cases. As a byproduct we show that every finitely generated abelian group appears as the class group of the local ring at the vertex of a cone over some smooth complex variety of each positive dimension.

math.AG↗

arXiv2016

Grothendieck-Lefschetz Theorem with Base Locus

We compute the divisor class group of the general hypersurface Y of a complex projective normal variety X of dimension at least four containing a fixed base locus Z. We deduce that completions of normal local complete intersection domains of finite type over the complex numbers of dimension $\ge 4$ are completions of UFDs of finite type over the complex numbers.

math.AG↗

arXiv2014

Srinivas' Problem for Rational Double Points

For the completion B of a local geometric normal domain, V. Srinivas asked which subgroups of Cl B arise as the image of the map from Cl A to Cl B on class groups as A varies among normal geometric domains with B isomorphic to the completion of A. For two dimensional rational double point singularities we show that all subgroups arise in this way. We also show that in any dimension, every normal hypersurface singularity has completion isomorphic to that of a geometric UFD. Our methods are global, applying Noether-Lefschetz theory to linear systems with non-reduced base loci.

math.AG↗

arXiv2012

Picard Groups of Normal Surfaces

We study the fixed singularities imposed on members of a linear system of surfaces in P^3_C by its base locus Z. For a 1-dimensional subscheme Z \subset P^3 with finitely many points p_i of embedding dimension three and d >> 0, we determine the nature of the singularities p_i \in S for general S in |H^0 (P^3, I_Z (d))| and give a method to compute the kernel of the restriction map Cl S \to Cl O_{S,p_i}. One tool developed is an algorithm to identify the type of an A_n singularity via its local equation. We illustrate the method for representative Z and use Noether-Lefschetz theory to compute Pic S.

math.AG↗

arXiv2011

Local Picard Groups

We use our extension of the Noether-Lefschetz theorem to describe generators of the class groups at the local rings of singularities of very general hypersurfaces containing a fixed base locus. We give several applications, including (1) every subgroup of the class group of the completed local ring of a rational double point arises as the class group of such a singularity on a surface in complex projective 3-space and (2) every complete local ring arising from a normal hypersurface singularity over the complex numbers is the completion of a unique factorization domain of essentially finite type over the complex numbers.

math.AG↗

arXiv2010

Detaching embedded points

We show that if $D \subset \mathbb P^N$ is obtained from a codimension two local complete intersection $C$ by adding embedded points of multiplicity $\leq 3$, then $D$ is a flat limit of $C$ and isolated points. As applications, we determine the irreducible components of Hilbert schemes of space curves with high arithmetic genus, show the smoothness of the Hilbert component whose general member is a plane curve union a point in $\mathbb P^3$, and construct a Hilbert component whose general member has an embedded point.

math.AG↗

arXiv2009

Noether-Lefschetz theorem with base locus

We compute the class groups of very general normal surfaces in complex projective three-space containing an arbitrary base locus $Z$, thereby extending the classic Noether-Lefschetz theorem (the 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, which generalize several known results.

math.AG↗

arXiv2009

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 > 2. 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.

math.AG↗

arXiv2007

Birationality of étale morphisms via surgery

We use a counting argument and surgery theory to show that if $D$ is a sufficiently general algebraic hypersurface in $\Bbb C^n$, then any local diffeomorphism $F:X \to \Bbb C^n$ of simply connected manifolds which is a $d$-sheeted cover away from $D$ has degree $d=1$ or $d=\infty$ (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 $F:X \to \Bbb C^n$ of algebraic varieties which covers away from such a hypersurface $D$ must be birational.

math.AG↗

arXiv2005

Holomorphic injectivity and the Hopf map

We give sharp conditions on a local biholomorphism $F:X \to \mathbb C^{n}$ which ensure global injectivity. For $n \geq 2$, such a map is injective if for each complex line $l \subset \mathbb C^{n}$, the pre-image $F^{-1}(l)$ embeds holomorphically as a connected domain into $\mathbb C \mathbb P^{1}$, the embedding being unique up to Möbius transformation. In particular, $F$ is injective if the pre-image of every complex line is connected and conformal to $\mathbb C$. The proof uses the topological fact that the natural map $\mathbb R \mathbb P^{2n-1} \to \mathbb C \mathbb P^{n-1}$ associated to the Hopf map admits no continuous sections and the classical Bieberbach-Gronwall estimates from complex analysis.

math.AG↗

arXiv2001

Hilbert Schemes of Degree Four Curves

In this paper we determine the irreducible components of the Hilbert schemes H(4,g) of locally Cohen-Macaulay space curves of degree four and arbitrary arithmetic genus g. We show that these Hilbert schemes are connected, in spite of having about g^2/24 irreducible components. For g < -2 we exhibit a component that is disjoint from the component of extremal curves and use this to give a counterexample to a conjecture of Ait-Amrane and Perrin.

math.AG↗

arXiv2001

Curves on a Double Surface

Let F be a smooth surface in a smooth projective threefold T, and let X=2F be the first infinitesimal neighborhood of X in T. A locally Cohen-Macaulay curve C in X gives rise to two effective divisors on F, namely the curve part P of the intersection of C and F, and the curve R residual in C to this intersection. We show that a general deformation of R on F lifts to a deformation of C on X when a certain cohomology group vanishes. In our paper "Hilbert Schemes of Degree Four Curves" we use this result to prove the connectedness of the Hilbert schemes H(4,g) of locally Cohen-Macaulay space curves of degree four and arbitrary arithmetic genus g.

math.AG↗

arXiv1996

The Hilbert Schemes of Degree Three Curves are Connected

In this paper we show that the Hilbert scheme $H(3,g)$ of locally Cohen-Macaulay curves in $\Pthree$ of degree three and genus $g$ is connected. In contrast to $H(2,g)$, which is irreducible, $H(3,g)$ generally has many irreducible components (roughly $-g/3$ of them). To show connectedness, we classify the curves (giving particular attention to the triple lines), determine the irreducible components, and give flat families over $\Aone$ to show that the components meet. As a byproduct, we find that there are curves which lie in the closure of each irreducible component.

alg-geom↗

arXiv1995

Integral Subschemes of Codimension Two

In this paper we study the problem of describing the integral subschemes within a fixed even linkage class $Ł$ of subschemes in $\Pn$ of codimension two. In the case that $Ł$ is not the class of arithmetically Cohen-Macaulay subschemes, we associate to any $X \in Ł$ two invariants $θ_X$ and $η_X$. When taken with the height $h_X$, each of these invariants determines the location of $X$ in $Ł$, thought of as a poset under domination. In terms of these invariants, necessary conditions are given for integral subschemes. The necessary conditions are almost sufficient in the sense that if a subscheme $X$ satisfies the necessary conditions and dominates an integral subscheme $Y$, then $X$ can be deformed with constant cohomology through subschemes in $Ł$ to an integral subscheme. In particular, if an even linkage class has a minimal element which is integral, then the conditions are both necessary and sufficient.

alg-geom↗