Linear Systems Of Plane Curves Research Articles

On the unique unexpected quartic in {mathbb {P}}^2

The computation of the dimension of linear systems of plane curves through a bunch of given multiple points is one of the most classic issues in algebraic geometry. In general, it is still an open problem to understand when the points fail to impose independent conditions. Despite many partial results, a complete solution is not known, even if the fixed points are in general position. The answer in the case of general points in the projective plane is predicted by the famous Segre–Harbourne–Gimigliano–Hirschowitz conjecture. When we consider fixed points in special position, even more interesting situations may occur. Recently, Di Gennaro, Ilardi and Vallès discovered a special configuration Z of nine points with a remarkable property: A general triple point always fails to impose independent conditions on the ideal of Z in degree four. The peculiar structure and properties of this kind of unexpected curves were studied by Cook II, Harbourne, Migliore and Nagel. By using both explicit geometric constructions and more abstract algebraic arguments, we classify low-degree unexpected curves. In particular, we prove that the aforementioned configuration Z is the unique one giving rise to an unexpected quartic.

Number of rational branches of a singular plane curve over a finite field

Let F be a plane singular curve defined over a finite field Fq. Via results of [11] and [1], the linear system of plane curves of a given degree passing through the singularities of F provides potentially good bounds for the number of points on a non-singular model of F. In this note, the case of a curve with two singularities such that the sum of their multiplicities is precisely the degree of the curve is investigated in more depth. In particular, such plane models are completely characterized, and for p>3, a curve of this type attaining one of the obtained bounds is presented.

Frobenius nonclassicality with respect to linear systems of curves of arbitrary degree

For each integer $s \geq 1$, we present a family of curves that are $\mathbb {F}_q$-Frobenius nonclassical with respect to the linear system of plane curves of degree $s$. In the case $s=2$, we give necessary and sufficient conditions for such curves to b

Homogeneous interpolation on ten points

In this paper we prove that for all pairs ( d , m ) (d,m) with d / m ≥ 174 / 55 d/m\geq 174/55 , the linear system of plane curves of degree d d with ten general base points of multiplicity m m has the expected dimension.

On a theorem of Castelnuovo and applications to moduli

In this paper we prove a theorem stated by Castelnuovo which bounds the dimension of linear systems of plane curves in terms of two invariants, one of which is the genus of the curves in the system. Then we classify linear systems whose dimension belongs to certain intervals which naturally arise from Castelnuovo's theorem. Finally we make an application to the following moduli problem: what is the maximum number of moduli of curves of geometric genus $g$ varying in a linear system on a surface? It turns out that, for $g\ge 22$, the answer is $2g+1$, and it is attained by trigonal canonical curves varying on a balanced rational normal scroll.

Characteristic 2 approach to bivariate interpolation problems

We investigate bivariate Hermite interpolation problems in characteristic 2. Given a nonnegative integer t, we describe all the sub-linear systems generated by monomials, in which there is no curve passing through a general point with multiplicity at least 2t. As an application, we show that a certain linear system of plane curves with ten base points is non-special.

Curves having one place at infinity and linear systems on rational surfaces

Denoting by L d ( m 0 , m 1 , … , m r ) the linear system of plane curves of degree d passing through r + 1 generic points p 0 , p 1 , … , p r of the projective plane with multiplicity m i (or larger) at each p i , we prove the Harbourne–Hirschowitz Conjecture for linear systems L d ( m 0 , m 1 , … , m r ) determined by a wide family of systems of multiplicities m = ( m i ) i = 0 r and arbitrary degree d . Moreover, we provide an algorithm for computing a bound for the regularity of an arbitrary system m , and we give its exact value when m is in the above family. To do that, we prove an H 1 -vanishing theorem for line bundles on surfaces associated with some pencils “at infinity”.

New effective bounds on the dimension of a linear system in [formula omitted

The main goal of this paper is to present an algorithm bounding the dimension of a linear system of plane curves of given degree (or monomial basis) with multiple points in general position. As a result we prove the Harbourne–Hirschowitz conjecture when the multiplicities of base points are bounded by 11. This gives a partial answer to the question of when bivariate polynomial interpolation is possible.

Nagata’s conjecture for a square or nearly-square number of points

In this paper we prove that a linear system of plane curves of degree d with n points of multiplicity m always has the expected dimension when n is a perfect square. We also prove some results on the dimension of the linear system when n is close to a perfect square. Keywords: Interpolation, Nagata, Linear systems, Plane curves Mathematics Subject Classification (2000): 14J26, 14N05

Linear systems of plane curves with a composite number of base points of equal multiplicity

In this article we study linear systems of plane curves of degree d d passing through general base points with the same multiplicity at each of them. These systems are known as homogeneous linear systems. We especially investigate for which of these systems, the base points, with their multiplicities, impose independent conditions and which homogeneous systems are empty. Such systems are called non-special. We extend the range of homogeneous linear systems that are known to be non-special. A theorem of Evain states that the systems of curves of degree d d with 4 h 4^h base points with equal multiplicity are non-special. The analogous result for 9 h 9^h points was conjectured. Both of these will follow, as corollaries, from the main theorem proved in this paper. Also, the case of 4 h 9 k 4^{h}9^{k} points will follow from our result. The proof uses a degeneration technique developed by C. Ciliberto and R. Miranda.

Box-shaped matrices and the defining ideal of certain blowup surfaces

In this paper, we generalize the notions of a matrix and its ideal of 2×2 minors to that of a box-shaped matrix and its ideal of 2×2 minors, and make use of these notions to study projective embeddings of certain blowup surfaces. We prove that the ideal of 2×2 minors of a generic box-shaped matrix is a perfect prime ideal that gives the algebraic description for the Segre embedding of the product of several projective spaces. We use the notion of the ideal of 2×2 minors of a box-shaped matrix to give an explicit description for the defining ideal of the blowup of P 2 along a set of ( d+1 2 ) (d∈ Z) points in generic position, embedded into projective spaces using very ample divisors which correspond to the linear systems of plane curves going through these points.

Linear systems of plane curves with imposed multiple points

A conjecture of Harbourne and Hirschowitz implies that $r \ge 9$ general points of multiplicity $m$ impose independent conditions to the linear system of curves of degree $d$ when $d(d+3) \ge rm(m+1)-2$. In this paper we prove that the conditions are independent provided $d+2\ge (m+1)(\sqrt{r+1.9}+\pi /8) $.

THE DIMENSION OF QUASIHOMOGENEOUS PLANAR LINEAR SYSTEMS WITH MULTIPLICITY FOUR

A linear system of plane curves satisfying multiplicity conditions at points in general position is called special if the dimension is larger than the expected dimension. A (−1) curve is an irreducible curve with self intersection −1 and genus zero. The Segre–Harbourne–Hirschowitz Conjecture states that if a linear system is special then a multiple of some fixed (−1) curve is contained in every curve of the linear system. This conjecture is proven for linear systems with multiplicity four at all but one of the points.

Linear systems of plane curves with base points of equal multiplicity

In this article we address the problem of computing the dimension of the space of plane curves of degree d d with n n general points of multiplicity m m . A conjecture of Harbourne and Hirschowitz implies that when d ≥ 3 m d \geq 3m , the dimension is equal to the expected dimension given by the Riemann-Roch Theorem. Also, systems for which the dimension is larger than expected should have a fixed part containing a multiple ( − 1 ) (-1) -curve. We reformulate this conjecture by explicitly listing those systems which have unexpected dimension. Then we use a degeneration technique developed to show that the conjecture holds for all m ≤ 12 m \leq 12 .

Calculs de dimensions de systèmes linéaires de courbes planes par collisions de gros points

What is the dimension of the linear system of plane curves of degree d going through points p 1,…, p r in general position with multiplicity m. 1,…, m r? In this Note, we introduce the notion of collision of fat points, and we explain that, in some sense, it is equivalent to understand collisions of fat points or to calculate this dimension.

Bott’s formula and enumerative geometry

We outline a strategy for computing intersection numbers on smooth varieties with torus actions using a residue formula of Bott. As an example, Gromov-Witten numbers of twisted cubic and elliptic quartic curves on some general complete intersection in projective space are computed. The results are consistent with predictions made from mirror symmetry computations. We also compute degrees of some loci in the linear system of plane curves of degrees less than 10, like those corresponding to sums of powers of linear forms, and curves carrying inscribed polygons.

Linear systems of plane curves through fixed “fat” points of [formula omitted

Given any s points P 1,…, P s in P 2 and s positive integers m 1,…, m s , let S n be the linear system of plane curves of degree n through P i with multiplicity at least m i (1 ⩽ i ⩽ s). We give numerical bounds for the regularity of S n in the following cases (a) the points P i are non-singular points of an integral curve of degree d; (b) the P i 's are in general position; (c) the P i 's are in uniform position; (d) the P i 's are generic points of P 2. We also study the sharpness of such bounds.

Regularity of linear systems of plane curves

The subject of this paper is the study of linear systems of plane curves which are given by assigning a set of fixed multiple points. This study not only has an interest per se, but is connected to other subjects (for example, results on rational surfaces and curves on them can be found in [Ro], [A], and [Gil). A linear system of plane curves of (projective) dimension r is a subvector space I, s

On singularities of the curves belonging to some linear system inP 2

In this paper we deal with singularities of the linear systems of plane curves passing through S, where S is a zerodimensional closed subscheme of degree n ofP2=P k 2 ,k an algebraically closed field of any characteristic. We determine the least degree of a nonsingular curve passing through S, when S is in uniform position.

On the Representation of Circles by means of Points in Space of Three Dimensions

The idea of the representation of a linear system of plane curves by means of a flat space is of fundamental importance in algebraic geometry. In this paper some details of the representation of circles are worked out, and an attempt is made to give a three-dimensional picture of certain parts of circle geometry The theorems given in italics are thought to be new, although in a subject with such an enormous literature it might almost be said that no theorem can be a new one.