On the Hilbert Function of General Unions of Curves in Projective Spaces

This paper is the second of a two-part series by the author devoted to the following fundamental problem in the theory of algebraic curves in projective space: Which reducible curves arise as limits of smooth curves of general moduli? Special cases of this question studied by Sernesi (Sernesi, Edoardo (1984) On the existence of certain families of curves. Invent Math 75(1): 488 25-57 ), Ballico (Ballico, Edoardo (2012) Embeddings of general curves in projective spaces: the range of the quadrics. Lith Math J 52(2): 134-137 ), and others have been critical in the resolution of many problems in the theory of algebraic curves over the past half century. In this paper, we give sharp bounds on this problem for space curves, when the nodes are general points and the components are general in moduli. We also systematically study a variant in projective spaces of arbitrary dimension when the nodes are general in a hyperplane. The results given here significantly extend those cases established in the previous paper in this series (Eric Larson, Constructing reducible Brill-Noether curves, To appear in documentamathematica, arxiv:1603.02301), as well as those cases established by Sernesi (Sernesi, Edoardo (1984) On the existence of certain families of curves. Invent Math 75(1): 488 25-57 ), Ballico (Ballico, Edoardo (2012) Embeddings of general curves in projective spaces: the range of the quadrics. Lith Math J 52(2): 134-137 ) , and others. As explained in (Eric Larson, Degenerations of curves in projective space and the maximal rank conjecture, arXiv:1809.05980), the reducible curves constructed in this paper also play a critical role in the author’s proof of the maximal rank conjecture in a subsequent paper (Eric Larson Degenerations of curves in projective space and the maximal rank conjecture, arXiv:1809.05980).

An efficient method based on projective joint invariant signatures is presented for distributed matching of curves in a camera network. The fundamental projective joint invariants for curves in the real projective space are the volume cross-ratios. A curve in m-dimensional projective space is represented by a signature manifold comprising n-point projective joint invariants, where n is at least m + 2. The signature manifold can be used to establish equivalence of two curves in projective space. However, without correspondence between the two curves, matching signature manifolds is a computational challenge. In this paper we overcome this challenge by finding discriminative sections of signature manifolds consistently across varying viewpoints and scoring the similarity between these sections. This motivates a simple yet powerful method for distributed curve matching in a camera network. Experimental results with real data demonstrate the classification performance of the proposed algorithm with respect to the size of the sections of the invariant signature in various noisy conditions.

A novel method is presented for distributed matching of curves across widely varying viewpoints. The fundamental projective joint-invariants for curves in the real projective space are the volume cross ratios. A curve in m-dimensional projective space is represented by a signature manifold comprising n-point projective joint invariants, where n is at least m+2. The signature manifold can be used to establish equivalence of two curves in projective space. However, without correspondence information, matching signature manifolds is a computational challenge. Our approach in this chapter is to first establish best possible correspondence between two curves using sections of the invariant signature manifold and then perform a simple test for equivalence. This allows fast computation and matching while keeping the descriptors compact. The correspondence and equivalence of curves is established independently at each camera node. Experimental results with simulated as well as real data are provided.

In this paper, we give a sufficient numerical criterion for a monomial curve in a projective space to be a set-theoretic complete intersection. Our main result generalizes a similar statement proven by Keum for monomial curves in three-dimensional projective space. We also prove that there are infinitely many set-theoretic complete intersection monomial curves in the projective n−space for any suitably chosen n − 1 integers. In particular, for any positive integers p, q, where gcd(p, q) = 1, the monomial curve defined by p, q, r is a set-theoretic complete intersection for every \({r \geq pq( q - 1)}\).

We discuss several questions on the geometry of curves in projective spaces: existence or non-existence for prescribed degrees and genera, their Hilbert function and their gonality.

The space of smooth genus- 0 0 curves in projective space has a natural smooth compactification: the moduli space of stable maps, which may be seen as the generalization of the classical space of complete conics. In arbitrary genus, no such natural smooth model is expected, as the space satisfies “Murphy’s Law”. In genus 1 1 , however, the situation remains beautiful. We give a natural smooth compactification of the space of elliptic curves in projective space, and describe some of its properties. This space is a blowup of the space of stable maps. It can be interpreted as a result of blowing up the most singular locus first, then the next most singular, and so on, but with a twist—these loci are often entire components of the moduli space. We give a number of applications in enumerative geometry and Gromov-Witten theory. For example, this space is used by the second author to prove physicists’ predictions for genus-1 Gromov-Witten invariants of a quintic threefold. The proof that this construction indeed gives a desingularization will appear in a subsequent paper.

(1989). Differential Invariants of Curves in Projective Space. Mathematics Magazine: Vol. 62, No. 1, pp. 28-35.

We consider the question: “How bad can the deformation space of an object be?” The answer seems to be: “Unless there is some a priori reason otherwise, the deformation space may be as bad as possible.” We show this for a number of important moduli spaces. More precisely, every singularity of finite type over ℤ (up to smooth parameters) appears on: the Hilbert scheme of curves in projective space; and the moduli spaces of smooth projective general-type surfaces (or higher-dimensional varieties), plane curves with nodes and cusps, stable sheaves, isolated threefold singularities, and more. The objects themselves are not pathological, and are in fact as nice as can be: the curves are smooth, the surfaces are automorphism-free and have very ample canonical bundle, the stable sheaves are torsion-free of rank 1, the singularities are normal and Cohen-Macaulay, etc. This justifies Mumford’s philosophy that even moduli spaces of well-behaved objects should be arbitrarily bad unless there is an a priori reason otherwise. Thus one can construct a smooth curve in projective space whose deformation space has any given number of components, each with any given singularity type, with any given non-reduced behavior. Similarly one can give a surface over $\mathbb{F}_{p}$ that lifts to ℤ/p 7 but not ℤ/p 8. (Of course the results hold in the holomorphic category as well.) It is usually difficult to compute deformation spaces directly from obstruction theories. We circumvent this by relating them to more tractable deformation spaces via smooth morphisms. The essential starting point is Mnev’s universality theorem.

A fundamental problem in the theory of algebraic curves in projective space is to understand which reducible curves arise as limits of smooth curves of general moduli. Special cases of this question and variants have been critical in the resolution of many problems in the theory of algebraic curves over the past half century; examples include Sernesi's proof of the existence of components of the Hilbert scheme with the expected number of moduli when the Brill–Noether number is negative [ E. Sernesi , Invent. Math. 75, 25–57 (1984; Zbl 0541.14024)], and Ballico's proof the Maximal Rank Conjecture for quadrics [ E. Ballico , Lith. Math. J. 52, No. 2, 134–137 (2012; Zbl 1282.14054)]. In this paper, we give close-to-optimal bounds on this problem when the nodes are general points and the components are general in moduli. The results given here significantly extend those cases established by [Sernesi, loc. cit.], [Ballico, loc. cit.], and others. As explained in [ Eric Larson , "Degenerations of curves in projective space and the maximal rank conjecture", Preprint (2018); ], they also play a key role in the author's proof of the Maximal Rank Conjecture [ Eric Larson , "The maximal rank conjecture", Preprint (2017); ].

Equivalence of variational problems of higher order

We present an approach to a large class of enumerative problems concerning rational curves in projective spaces. This approach uses analysis to obtain topological information about moduli spaces of stable maps. We demonstrate it by enumerating one-component rational curves with a triple point or a tacnodal point in the three-dimensional projective space and with a cusp in any projective space.

In studying algebraic curves in projective spaces, our forefathers in the 19th century noted that curves naturally move in algebraic families. In the projective plane, this is a simple matter. A curve of degree d is defined by a single homogeneous polynomial in the homogeneous coordinates X 0,x 1,X 2. The coefficients of this polynomial give a point in another projective space, and in this way curves of degree d in the plane are parametrized by the points of a ℙN with \( N = \tfrac{1} {2}d(d + 3) \). For an open set of ℙN, the corresponding curve is irreducible and nonsingular. The remaining points of ℙN correspond to curves that are singular, or reducible, or have multiple components. In particular, the nonsingular curves of degree d in ℙ2 form a single irreducible family.

We determine the splitting (isomorphism) type of the normal bundle of a generic genus-0 curve with 1 or 2 components in any projective space, as well as the (sometimes nontrivial) way the bundle deforms locally with a general deformation of the curve. We deduce an enumerative formula for divisorial loci of smooth rational curves whose normal bundle is of non-generic splitting type.

We use explicit results on modular forms (Muic, Ramanujan J 27:188–208, 2012) via uniformization theory to obtain embeddings of modular curves and more generally of compact Riemann surfaces attached to Fuchsian groups of the first kind in certain projective spaces. We obtain families of embeddings which vary smoothly with respect to a parameter in the upper-half plane. We study local expression for the divisors attached to the maps in the family.

We study an isomorphism between the group of rigid body displacements and the group of dual quaternions modulo the dual number multiplicative group from the viewpoint of differential geometry in a projective space over the dual numbers. Some seemingly weird phenomena in this space have lucid kinematic interpretations. An example is the existence of non-straight curves with a continuum of osculating tangents which correspond to motions in a cylinder group with osculating vertical Darboux motions. We also look at the set of osculating conics of a curve in projective space, suggest geometrically meaningful examples and briefly discuss and illustrate their corresponding motions.