Rational Quartic Curve Research Articles

Reconstructing curves from their Hodge classes

Let S be a smooth algebraic surface in {mathbb {P}}^3({mathbb {C}}). Movasati and Sertöz (Rend. Circ. Mat. Palermo 2:1–17, 2020) associate an ideal I_{alpha (C)} to the primitive cohomology class alpha (C) of C in S. We show that the equations of C can be determined by I_{alpha (C)} under numerical conditions. We apply this result to reconstruct rational curves and arithmetically Cohen-Macaulay curves from their cohomology classes. On the other hand, we show that the class alpha (C) of a rational quartic curve C on a smooth quartic surface S is not even perfect, that is, that I_{alpha (C)} is bigger than the sum of the Jacobian ideal of S and of the homogeneous ideals of curves D in S for which I_{alpha (D)}=I_{alpha (C)}.

Moduli of sheaves supported on Quartic space curves

As a continuation of the work of Freiermuth and Trautmann, we study the geometry of the moduli space of stable sheaves on $\mathbb{P}^3$ with Hilbert polynomial $4m+1$. The moduli space has three irreducible components whose generic elements are, respectively, sheaves supported on rational quartic curves, on elliptic quartic curves, or on planar quartic curves. The main idea of the proof is to relate the moduli space with the Hilbert scheme of curves by wall crossing. We present all stable sheaves contained in the intersections of the three irreducible components. We also classify stable sheaves by means of their free resolutions.

Existence and computation of spherical rational quartic curves for Hermite interpolation

Existence and computation of spherical rational quartic curves for Hermite interpolation

A symmetrical configuration of n + 1 rational normal curves in [2n

This paper is concerned with the extension to [2n] of a well-known symmetrical configuration in [4], namely, that of three rational quartic curves, with six points in common, having the property that a trisecant plane of any one which meets a second also meets the third.

Book Review: The Rational Quartic Curve in Space of Three and Four Dimensions

Book Review: The Rational Quartic Curve in Space of Three and Four Dimensions

The Rational Quartic Curve in Space of Three and Four Dimensions. By H. G. Telling. Pp. viii, 78. 5s. 1936. Cambridge Tracts, 34. (Cambridge)

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The Rational Quartic Curve in Space of Three and Four Dimensions

IN this tract Miss Telling has presented in compact form a great deal of information, gathered from various sources, which is not available as a whole elsewhere. The tract is divided into two chapters and a short appendix. The first chapter deals with the four-dimensional curve and its projective generation, invariants and the symbolical notation for them, fundamental polarity, trisecant planes, chords, quadratic involutions, director lines and planes, (/-lines and the manifold G, the chordal J, the quartic surface K and its nodes, Segre cubic primals, and linear complexes containing the curve, or apolar to it. In the second chapter we descend to three-dimensional space, and consider quartic curves of the first and second kind, principal, quadratic, and cubic involutions; flexes, trisecants and Hessian points, the surfaces of Steiner and Veronese, and several special kinds of quartic curves. The appendix consists of a note on involutions on the four-dimensional quartic. The Rational Quartic Curve in Space of Three and Four Dimensions: being an Introduction to Rational Curves. By H. G. Telling. (Cambridge Tracts in Mathematics and Mathematical Physics, No. 34.) Pp. viii + 78. (Cambridge: At the University Press, 1936.) 5s. net.

Related quadrics and systems of a rational quartic curve

1·1. The points, tangents, osculating planes, …, osculating primes of a curve may be said to form a system which is characterised by the number of these elements which are incident with a prime, …, line, point, respectively. For the normal rational quartic curve the system is (4, 6, 6, 4); projection of this system from a general point gives the system (4, 6, 6) in [3]; section by a general prime gives the system (6, 6, 4). These two systems in [3], which are the systems with which we are concerned in this paper, are duals of one another, and will be called systems of the first and second kinds respectively.

The Number of Rational Quartic Curves with Seven Assigned Trisecant Planes in Space of Four Dimensions

Journal of the London Mathematical SocietyVolume s1-8, Issue 1 p. 9-11 Notes and Papers The Number of Rational Quartic Curves with Seven Assigned Trisecant Planes in Space of Four Dimensions D. W. Babbage, D. W. BabbageSearch for more papers by this author D. W. Babbage, D. W. BabbageSearch for more papers by this author First published: January 1933 https://doi.org/10.1112/jlms/s1-8.1.9AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat No abstract is available for this article. Volumes1-8, Issue1January 1933Pages 9-11 RelatedInformation

Plane congruences of the second order in space of four dimensions and fifth incidence theorems

It is well known that the planes which meet four given lines in a space of four dimensions meet a fifth line, determined by the first four. Also the trisecant planes of a rational quartic curve in [4] which meet a line meet another rational quartic curve. These two theorems are of the same type and may conveniently be called “fifth incidence theorems”.

Some enumerative results for curves

1. The problem of finding the number of space cubic curves which pass through p given points and have 6 – p given lines as chords has been solved by several different methods. The similar problem, in space of four dimensions, of the number of rational quartic curves which pass through p given points and have 7 – p given trisecant planes has been solved for p > 1 by F. P. White and for p = 1 by J. A. Todd. In a recent paper I discussed a similar problem for elliptic quartic curves. My present object is to apply the method of that paper to the problems mentioned above and to obtain two other sets of numbers which I believe to be new. They are (1) the number of rational normal quartic curves which have three assigned chords, pass through p assigned points, and have 3 – p assigned trisecant planes; (2) the number of curves of intersection of three quadrics in [4] which have a suitable number of assigned points and chords. The evaluation of Schubert symbols which occur in the work is done by means of a formula due to Giambelli‖. This formula is stated in a more simple form in a note at the end of this paper (7).

Rational Tacnodal and Oscnodal Quartic Curves Considered as Plane Sections of Quartic Surfaces

Introduction. It is well known that the plane sections of the Steiner Quartic Surface include all types to within projection of the rational plane quartic curve with simple singularities and that the invariants of any curve cut from this surface may be expressed as symmetric functions of the coefficients of the cutting plane.f If we vary this surface so that two of its double lines become coincident the plane sections are rational quartic curves with tacnodes; t if the three double lines are made coincident the resulting surface has as plane sections rational quartic curves with osenodes.? It is the purpose of this paper to express the invariants of the plane sections of these two surfaces in terms of the coefficients of the cutting plane.

On a Type of Plane Rational Quartic Curve

On a Type of Plane Rational Quartic Curve

Important Covariant Curves and a Complete System of Invariants of the Rational Quartic Curve

Important Covariant Curves and a Complete System of Invariants of the Rational Quartic Curve

Important covariant curves and a complete system of invariants of the rational quartic curve

It is well understood that different domains of rationality are useful ill discussing different properties of curves. Two domains are employed in the folSowing, to render possible the geometric interpretation of certain invariants and covariant loci of the rational plane quartic. Section 1 is divided into two parts: In the first part is given a straightfozward proof of the covariance of curves derived from Xn by a certain translation scheme; in the second part SALMONES work on the combinants of two binary quartics is applied to those covariant curves of the R4 which can be found as combinants. In Section 2 the most important invariants of the Rw are discussed, and four invariants are found in terms of which any other invariant relation on the R can be expressed algebraically; in this sense these four invariants constitute an algebraically complete system. Section 3 contains a treatment of the invariants of the 4 wllen the R4 is taken as the section of the Steiner Quartic Surface by a plane; in this schenle the invariants occur as symmetric functions of the coefficients of the cutting plane.

The Osculants of Plane Rational Quartic Curves

The Osculants of Plane Rational Quartic Curves