Ordinary Double Points Research Articles

Degeneration of trigonal curves and solutions of the KP-hierarchy

It is known that soliton solutions of the KP-hierarchy correspond to singular rational curves with only ordinary double points. In this paper we study the degeneration of theta function solutions corresponding to certain trigonal curves. We show that, when the curves degenerate to singular rational curves with only ordinary triple points, the solutions tend to be some intermediate solutions between solitons and rational solutions. They are considered as certain limits of solitons.The Sato Grassmannian is extensively used here to study the degeneration of solutions, since it directly connects solutions of the KP-hierarchy to the defining equations of algebraic curves. We define a class of solutions in the Wronskian form which contain soliton solutions as a subclass and prove that, using the Sato Grassmannian, the degenerate trigonal solutions are connected to those solutions by certain gauge transformations.

Classification of real rational knots of low degree in the 3-sphere

In this paper, we classify, up to rigid isotopy, real rational knots of degrees less than or equal to [Formula: see text] in a real quadric homeomorphic to the 3-sphere. We also study their connections with rigid isotopy classes of real rational knots of low degree in [Formula: see text] and classify real rational curves of degree 6 in the 3-sphere with exactly one ordinary double point.

Liftability of Singularities and Their Frobenius Morphism Modulo $p^2$

We investigate the $W_2(k)$-liftability of singular schemes. We prove constructibility of the locus of $W_2(k)$-liftable schemes in a flat family $X \to S$. Moreover, we construct an explicit $W_2(k)$-lifting of a Frobenius split scheme $X$ over a perfect field $k$, reproving Bhatt's existential result. Furthermore, we study existence of liftings of the Frobenius morphism. In particular, we prove that in dimension $n \geq 4$ ordinary double points do not admit a $W_2(k)$-lifting compatible with Frobenius, and that canonical surface singularities are Frobenius liftable. Combined with Bhatt's results, the latter result implies that the crystalline cohomology groups over $k$ of surfaces with canonical singularities are not finite dimensional. As a corollary of our results, we provide a thorough comparison between the notions of $W_2(k)$-liftability, Frobenius liftability and classical $F$-singularity types.

On Plane Curves with Double and Triple Points

We describe in simple geometric terms the Hodge filtration on the cohomology $H^*(U)$ of the complement $U=\mathsf{P}^2 \setminus C$ of a plane curve $C$ with ordinary double and triple points. Relations to Milnor algebra, syzygies of the Jacobian ideal and pole order filtration on $H^2(U)$ are given.

Double point-homogeneous spherical curves

A curve is, in this paper, the image of the circle $S^1$ under an immersion $f$ into $S^2$, $\mathbb{R}^2$ or the real projective plane $P_2(\mathbb{R})$, such that every multiple point of $f$ is an ordinary double point. Such a curve $C$ is double point-homogeneous or DP-homogeneous when the group of diffeomorphisms (of $S^2, \mathbb{R}^2$ or $P_2(\mathbb{R})$) preserving $C$ has a transitive action on the set of its double points. The orbits of DP-homogeneous curves in $S^2$ are totally determined; using combinatorial methods, we prove that they fall into five countably infinite families ; the description of every family is illustrated by drawings of some representatives with a small number of double points. As a corollary, we obtain a similar classification of the DP-homogeneous curves in $\mathbb{R}^2$. We also propose a conjecture about the classification of DP-homogeneous curves in $P_2(\mathbb{R})$.

On a Conjecture of Candelas and de la Ossa

We prove that the metric completion of a canonical Ricci-flat Kahler metric on the nonsingular part of a projective Calabi-Yau variety $X$ with ordinary double point singularities, is a compact metric length space homeomorphic to the projective variety $X$ itself. As an application, we prove a conjecture of Candelas and de la Ossa for conifold flops and transitions.

Singularities of theta divisors and the geometry of $\mathcal A_5$

We study the codimension two locus H in \mathcal A_g consisting of principally polarized abelian varieties whose theta divisor has a singularity that is not an ordinary double point. We compute the class [H]\in CH^2(\mathcal A_g) for every g . For g=4 , this turns out to be the locus of Jacobians with a vanishing theta-null. For g=5 , via the Prym map we show that H\subset \mathcal A_5 has two components, both unirational, which we describe completely. We then determine the slope of the effective cone of \overline{\mathcal A_5} and show that the component \overline{N_0'} of the Andreotti-Mayer divisor has minimal slope and the Iitaka dimension \kappa(\overline{\mathcal A_5}, \overline{N_0'}) is equal to zero.

Torsion in the crystalline cohomology of singular varieties

This paper discusses some examples showing that the crystalline cohomology of even very mildly singular projective varieties tends to be quite large. In particular, any singular projective variety with at worst ordinary double points has infinitely generated crystalline cohomology in at least two cohomological degrees. These calculations rely critically on comparisons between crystalline and derived de Rham cohomology.

On a family of complex algebraic surfaces of degree 3n

We study a class of algebraic surfaces of degree 3n in the complex projective space with only ordinary double points. They are obtained by using bivariate polynomials with complex coefficients related to the generalized cosine associated with the affine Weyl group of the root system A2.

A closer look at the stacks of stable pointed curves

In the theory of the moduli-stacks of stable n-pointed curves, there are two fundamental functors, contraction and stabilization. These functors are constructed in Knudsen (1983) [3], where they are used to show that the various M¯g,n’s are DM-stacks. In the treatment of stabilization, there is a claim stated about the deformations of a pointed ordinary double point. This claim is not proved in Knudsen (1983) [3]. The goal of this paper is to prove this claim. We decided not to use descent, so the equations are less simple than in Knudsen (1983) [3], but the treatment is more elementary and the results seemingly more general.

A splitting criterion for an isolated singularity at x = 0 in a family of even hypersurfaces

Assume given a family of even local analytic hypersurfaces, whose central fiber has an isolated singularity at x = 0 which is not an ordinary double point. We prove that if the family is sufficiently general, for instance if the general fiber is smooth and the general singular fiber has only ordinary double points, then the singularity at x = 0 “splits in codimension one”, i.e., the local discriminant divisor has an irreducible component, over which a general fiber has more than one singularity specializing to the original one. As a corollary, we deduce the result by Grushevsky and Salvati Manni (Singularities of the theta divisor at points of order two, IMRN, 2007, Proposition 8) that on a principally polarized abelian variety (A, Θ) with dim(A) = g ≥ 4, a singularity of even multiplicity on Θ, isolated or not, at a point of order two and not an ordinary double point, must be a limit of two distinct ordinary double points {x, −x} on nearby theta divisors.

On a conjecture of Ciliberto

We prove that a threefold hypersurface of degree with at most ordinary double points is factorial if it contains no planes and has at most singular points. Bibliography: 13 titles.

Singularities of the Theta Divisor at Points of Order Two

In this note we study the geometry of principally polarized abelian varieties (ppavs) with a vanishing theta-null (i.e. with a singular point of order two and even multiplicity lying on the theta divisor). We describe the locus within the theta-null divisor where this singularity is not an ordinary double point. By using theta function methods we first show that this locus does not equal the entire theta-null divisor (this was shown previously by O. Debarre). We then show that this locus is contained in the intersection of the theta-null divisor with the other irreducible components of the Andreotti-Mayer divisor N_0, and describe by using the geometry of the universal scheme of singularities of the theta divisor the components of this intersection that are contained in this locus. Some of the intermediate results obtained along the way of our proof were concurrently obtained independently by C. Ciliberto and G. van der Geer, and by R. de Jong.

On the α-invariants of cubic surfaces with Eckardt points

In this paper, we show that the α m , 2 -invariant (introduced by Tian (1991) [27] and (1997) [29]) of a smooth cubic surface with Eckardt points is strictly bigger than 2 3 . This can be used to simplify Tian's original proof of the existence of Kähler–Einstein metrics on such manifolds. We also sketch the computations on cubic surfaces with one ordinary double points, and outline the analytic difficulties to prove the existence of orbifold Kähler–Einstein metrics.

Numerical properties of isotrivial fibrations

In this paper we investigate the numerical properties of relatively minimal isotrivial fibrations $\varphi \colon X \lr C$, where $X$ is a smooth, projective surface and $C$ is a curve. In particular we prove that, if $g(C) \geq 1$ and $X$ is neither ruled nor isomorphic to a quasi-bundle, then $K_X^2 \leq 8 \chi(\mO_X)-2$; this inequality is sharp and if equality holds then $X$ is a minimal surface of general type whose canonical model has precisely two ordinary double points as singularities. Under the further assumption that $K_X$ is ample, we obtain $K_X^2 \leq 8 \chi(\mO_X)-5$ and the inequality is also sharp. This improves previous results of Serrano and Tan.

Small resolutions and non-liftable Calabi-Yau threefolds

We use properties of small resolutions of the ordinary double point in dimension three to construct smooth non-liftable Calabi-Yau threefolds. In particular, we construct a smooth projective Calabi-Yau threefold over $${\mathbb{F}_3}$$ that does not lift to characteristic zero and a smooth projective Calabi-Yau threefold over $${\mathbb{F}_5}$$ having an obstructed deformation. We also construct many examples of smooth Calabi-Yau algebraic spaces over $${\mathbb{F}_p}$$ that do not lift to algebraic spaces in characteristic zero.

Factorial threefold hypersurfaces

Let X X be a hypersurface in P 4 \mathbb {P}^{4} of degree d d that has at worst isolated ordinary double points. We prove that X X is factorial in the case when X X has at most ( d − 1 ) 2 − 1 (d-1)^{2}-1 singular points.

Smoothing nodal Calabi-Yau n -folds

Let X be an n-dimensional Calabi-Yau with ordinary double points, where n is odd. Friedman showed that for n=3 the existence of a smoothing of X implies a specific type of relation between homology classes on a resolution of X. (The converse is also true, due to work of Friedman, Kawamata and Tian.) We sketch a more topological proof of this result, and then extend it to higher dimensions. For n>3 the "Yukawa product" on the middle dimensional (co)homology plays an unexpected role. We also discuss a converse, proving it for nodal Calabi-Yau hypersurfaces in projective space.

Non-commutative Donaldson–Thomas invariants and the conifold

Given a quiver algebra A with relations defined by a superpotential, this paper defines a set of invariants of A counting framed cyclic A–modules, analogous to rank–1 Donaldson–Thomas invariants of Calabi–Yau threefolds. For the special case when A is the non-commutative crepant resolution of the threefold ordinary double point, it is proved using torus localization that the invariants count certain pyramid-shaped partition-like configurations, or equivalently infinite dimer configurations in the square dimer model with a fixed boundary condition. The resulting partition function admits an infinite product expansion, which factorizes into the rank–1 Donaldson–Thomas partition functions of the commutative crepant resolution of the singularity and its flop. The different partition functions are speculatively interpreted as counting stable objects in the derived category of A–modules under different stability conditions; their relationship should then be an instance of wall crossing in the space of stability conditions on this triangulated category.

DESINGULARIZATIONS OF CALABI-YAU 3-FOLDS WITH CONICAL SINGULARITIES. II. THE OBSTRUCTED CASE

This is the second of two papers studying Calabi–Yau 3-folds with conical singularities and their desingularizations. In our first paper [Y.-M. Chan, Quart. J. Math. 57 (2006), 151–181] we constructed the desingularization of the conically singular manifold M0 by gluing an asymptotically conical (AC) Calabi–Yau 3-fold Y into M0 at the singular point, thus obtaining a 1-parameter family of compact, non-singular Calabi–Yau 3-folds Mt for small t > 0. During the gluing process one may encounter a kind of cohomological obstruction to defining a 3-form Ωt on Mt which interpolates between the 3-form Ω0 on M0 and the scaled 3-form t3 ΩY on Y if the rate λ at which the AC Calabi–Yau 3-fold Y converges to the Calabi–Yau cone is equal to − 3. The first paper [3] studied the simpler case λ < −3 where there is no obstruction. This paper extends the result in the first one by considering a more complicated situtation when λ = −3. Assuming the existence of singular Calabi–Yau metrics on compact complex 3-folds with ordinary double points, our result in this paper can be applied to repairing such kinds of singularities, which is an analytic version of Friedman's result giving necessary and sufficient conditions for smoothing ordinary double points.