The minimal resolution property for points on general curves

Several improvements to the computation of the minimal free resolution of finite modules have been made also recently ([5], [15]) and some of them concerning ideals of polynomials are Hilbert driven or depend on the knowledge of the regularity of the ideal. Here I show that some of the calculations usually made to determine the minimal free resolution of a homogeneous polynomial ideal I are still redundant, provided that we know a set of minimal homogeneous generators, the regularity, and the Hilbert function of I. More precisely, I show that the construction of some syzygies can be avoided. As result, the known methods for the computation of a minimal free resolution are improved. The underlying idea of this work can be also used to predict if certain points in generic position on general rational curves satisfy the minimal resolution conjecture.

Twisting of properads

While the Chebyshev pseudospectral method provides a spectrally accurate method, integration of partial differential equations with spatial derivatives of order $M$ requires time steps of approximately $O(N^{-2M})$ for stable explicit solvers. Theoretically, time steps may be increased to $O(N^{-M})$ with the use of a parameter, $\alpha$-dependent mapped method introduced by Kosloff and Tal-Ezer [{\em J. Comput. Phys}., 104 (1993), pp. 457--469]. Our analysis focuses on the utilization of this method for reasonable practical choices for $N$, namely $N \lesssim 30$, as may be needed for two- or three-dimensional modeling. Results presented confirm that spectral accuracy with increasing $N$ is possible both for constant $\alpha$ (Hesthaven, Dinesen, and Lynov [{\em J. Comput. Phys}., 155 (1999), pp. 287--306]) and for $\alpha$ scaled with $N$, $\alpha$ sufficiently different from $1$ (Don and Solomonoff [{\em SIAM J. Sci. Comput}., 18 (1997), pp. 1040--1055]). Theoretical bounds, however, show that any realistic choice for $\alpha$, in which both resolution and accuracy considerations are imposed, permits no more than a doubling of the time step for a stable explicit integrator in time, much less than the $O(N)$ improvement claimed by Kosloff and Tal-Ezer. On the other hand, by choosing $\alpha$ carefully, it is possible to improve on the resolution of the Chebyshev method; in particular, one may achieve satisfactory resolution with fewer than $\pi$ points per wavelength. Moreover, this improvement is noted not only for waves with the minimal resolution but also for waves sampled up to about $8$ points per wavelength. Our conclusions are verified by calculation of phase and amplitude errors for numerical solutions of first and second order one-dimensional wave equations. Specifically, while $\alpha$ can be chosen such that the mapped method improves the accuracy and resolution of the Chebyshev method, for practical choices of $N$, it is not possible to achieve both single precision accuracy and gain the advantage of an $O(N^{-M})$ time step.

A mixed quasi-étale quotient is the quotient of the product of a curve of genus at least 2 with itself by the action of a group which exchanges the two factors and acts freely outside a finite subset. A mixed quasi-étale surface is the minimal resolution of its singularities. We produce an algorithm computing all mixed quasi-étale surfaces with given geometric genus, irregularity and self-intersection of the canonical class. We prove that all irregular mixed quasi-étale surfaces of general type are minimal. As an application, we classify all irregular mixed quasi-étale surfaces of general type with genus equal to the irregularity, and all the regular ones with K2 > 0, thus constructing new examples of surfaces of general type with χ = 1. We mention the first example of a minimal surface of general type with pg = q = 1 and Albanese fibre of genus bigger than K2.

It is shown that the fixed part of the canonical linear system of a fibre in a relatively minimal fibred surface supports at most exceptional sets of weakly elliptic singular- ities. Introduction. Let S be a non-singular projective surface and f : S → C a surjective morphism of S onto a non-singular projective curve C with connected fibres. We call f a relatively minimal fibration of genus g if a general fibre is a non-singular projective curve of genus g and there are no (−1)-curves contained in fibres. We assume that g ≥ 2 throughout the paper. Let F be afi bre off . Then the intersection form is negative semi-definite on Supp(F ) by Zariski's lemma. Furthermore, there exist a positive integer m and a 1-connected curve D such that F = mD .W henm is strictly greater than one, F is called a multiple fibre of multiplicity m and OD(D) is a torsion of order m. In (8), we considered the canonical linear system on the minimal resolution of a normal surface singularity and showed that the fixed part supports at most exceptional sets of rational singular points (cf. (1) and (2)). The present article is an extension of it to the semi-global case and we study the fixed part of the canonical linear system |KF | which we call the canonical fixed part in this paper. Recall that the canonical fixed part is closely related to the Horikawa index (see (3, p. 12)), an analytic invariant of a singular fibre germ. In fact, according to (6, Lemma 10 and Theorem 3), if g = 2, the canonical fixed part is a chain of (−2)-curves (of type A) and the Horikawa index is almost equivalent to the number of its irreducible components.

Let S be any smooth toric surface. We establish a ring isomorphism between the equivariant extended Chen-Ruan cohomology of the n-fold symmetric product stack [Symn(S)] of S and the equivariant extremal quantum cohomology of the Hilbert scheme Hilbn(S) of n points in S. This proves a generalization of Ruan's Cohomological Crepant Resolution Conjecture for the case of Symn(S). Moreover, we determine the operators of small quantum multiplication by divisor classes on the orbifold quantum cohomology of [Symn(Ar)], where Ar is the minimal resolution of the cyclic quotient singularity C2/Zr+1. Under the assumption of the nonderogatory conjecture, these operators completely determine the quantum ring structure, which gives an affirmative answer to Bryan-Graber's Crepant Resolution Conjecture on [Symn(Ar)] and Hilbn(Ar). More strikingly, this allows us to complete a tetrahedron of equivalences relating the Gromov-Witten theories of [Symn(Ar)]/Hilbn(Ar) and the relative Gromov-Witten/Donaldson-Thomas theories of Ar x P1. Finally, we prove a closed formula for an excess integral over the moduli space of degree d stable maps from unmarked curves of genus one to the projective space Pr for positive integers r and d. The result generalizes the multiple cover formula for Pr and reveals that any simple Pr flop of smooth projective varieties preserves the theory of extremal Gromov-Witten invariants of arbitrary genus. It also provides examples for which Ruan's Minimal Model Conjecture holds.