Special Pencil Research Articles

High-school algebra of the theory of dicritical divisors: Atypical fibers for special pencils and polynomials

In this work we deal with dicritical divisors, curvettes and polynomials. These objects have been one of the main research interests of Abhyankar during his last years. In this work we provide some elementary proofs of some Abhyankar and Luengo results for dicriticals in the framework of formal power series. Based on these ideas we give a constructive way to find the atypical fibers of a special pencil and give bounds for its number, which are sharper than the existing ones. Finally, we answer a question of Gwoździewicz finding polynomials that reach his bound.

Syzygies of 5-gonal canonical curves

We show that for 5-gonal curves of odd genus g\ge 13 and even genus g\ge 28 the \lceil \frac{g-1}2\rceil -th syzygy module of the curve is not determined by the syzygies of the scroll swept out by the special pencil of degree 5.

Towards a tropical proof of the Gieseker–Petri Theorem

We use tropical techniques to prove a case of the Gieseker–Petri Theorem. Specifically, we show that the general curve of arbitrary genus does not admit a Gieseker–Petri special pencil.

Existence of dicritical divisors revisited

We characterize the dicriticals of special pencils. We also initiate higher dimensional dicritical theory.

More about dicriticals

This paper attempts to characterize the dicritical set of a rational function which generates a special pencil at a simple point of an algebraic or arithmetical surface.

Generalization of the potential method for blending three surfaces

The potential method is one of the methods of generating a blending surface along the intersection curves of implicit surfaces. The paper describes a generalized method, for three surfaces, of yielding the formulae for the convex and nonconvex combinations, each of which retains the locality of the blending regions, and has the ability to control the ranges of the edge blends along the original sharp edges. This is analogous to variable-radius rolling-ball blends. To obtain the satisfactory blending functions, the polarity and special pencils of quadrics are applied. After these derivations, it is pointed out that the other quadratic blends for three surfaces correspond to the special cases of the described method. Further, the projective potential method is presented for a convex combination of the three surfaces.

Unobtrusive Measures in Organizational Theory: A Reminder

You know, I am really stupid. For years I have looked for the perfect pencil. I have found very good ones, but never the perfect one. And all the time it was not the pencils but me. A pencil that is all right some days is no good another day. For example, yesterday I used a special pencil soft and fine and it floated over the paper just wonderfully. So this morning I tried the same kind. And they crack on me. Points break and all hell is let loose. This is the day when I am stabbing the paper. So today I need a harder pencil at least for a while. I am using some that are #2 & %. I have my plastic tray you know and in it three kinds of pencils for hard writing days and soft writing days. Only sometimes it changes in the middle of the day, but at least I am equipped for it. I have also some super-soft pencils which I do not use very often because I must feel as delicate as a rose petal to use them. And I am not often that way. But when I do have such moments I am prepared. It is always well to be prepared. Pencils are a great expense to me and I hope you know it. I buy them four dozen at a time. When in my normal writing position the metal of the pencil eraser touches my hand, I retire that pencil (Steinbeck, 1969: 35-36).

A remark on a property of a special pencil of quadrics

A remark on a property of a special pencil of quadrics

The Involution Curve Determined From a Special Pencil of n-ICS

The Involution Curve Determined From a Special Pencil of n-ICS

The involution curve determined from a special pencil of 𝑛-ics

The involution curve determined from a special pencil of 𝑛-ics

A special Pencil of Binary Quartics

I. If the Jacobian sextic of a pencil of binary quartics is known, the pencil itself is determined in five ways. The explicit determination of the pencil in terms of the irrational invariants of has been effected by Stephanos.There are two known cases in which a pencil of quartics admits of rational determination from its Jacobian. In these, the rationally determinable pencil differentiates itself algebraically from its remaining four co-Jacobian pencils.

Bhabha scattering and a special pencil of K3 surfaces

We study a pencil of K3 surfaces that appeared in the $2$-loop diagrams in Bhabha scattering. By analysing in detail the Picard lattice of the general and special members of the pencil, we identify the pencil with the celebrated Apery--Fermi pencil, that was related to Apery's proof of the irrationality of $\zeta(3)$ through the work of F. Beukers, C. Peters and J. Stienstra. The same pencil appears miraculously in different and seemingly unrelated physical contexts.