Let $E\subseteq \mathbb{P}^2$ be a complex rational cuspidal curve contained in the projective plane. The Coolidge-Nagata conjecture asserts that $E$ is Cremona equivalent to a line, i.e. it is mapped onto a line by some birational transformation of $\mathbb{P}^2$. In arXiv:1405.5917 the second author analyzed the log minimal model program run for the pair $(X,\frac{1}{2}D)$, where $(X,D)\to (\mathbb{P}^2,E)$ is a minimal resolution of singularities, and as a corollary he established the conjecture in case when more than one irreducible curve in $\mathbb{P}^2\setminus E$ is contracted by the process of minimalization. We prove the conjecture in the remaining cases.
Let E be a plane rational curve defined over complex numbers which has only locally irreducible singularities. The Coolidge-Nagata conjecture states that E is rectifiable, i.e. it can be transformed into a line by a birational automorphism of the plane. We show that if it is not rectifiable then the tree of the exceptional divisor for its minimal embedded resolution of singularities has at most nine maximal twigs. This settles the conjecture in case E has more than four singular points.
Let $E\subseteq \mathbb{P}^2$ be a complex rational cuspidal curve contained in the projective plane and let $(X,D)\to (\mathbb{P}^2,E)$ be the minimal log resolution of singularities. Applying the log minimal model program to $(X,\frac{1}{2}D)$ we prove that if $E$ has more than two singular points or if $D$, which is a tree of rational curves, has more than six maximal twigs or if $\mathbb{P}^2\setminus E$ is not of log general type then $E$ is Cremona equivalent to a line, i.e. the Coolidge-Nagata conjecture for $E$ holds. We show also that if $E$ is not Cremona equivalent to a line then th…
The Nagata Conjecture is one of the most intriguing open problems in the area of curves in the plane. It is easily stated. Namely, it predicts that the smallest degree d of a plane curve passing through r $\ge$ 10 general points in the projective plane P 2 with multiplicities at least l at every point, satisfies the inequality d > $\sqrt$ r $\times$ l. This conjecture has been proven by M. Nagata in 1959, if r is a perfect square greater than 9. Up to now, it remains open for every non-square r $\ge$ 10, after more than a half century of attention by many researchers. In this paper, we formula…
In this paper we discuss some variations of Nagata's conjecture on linear systems of plane curves. The most relevant concerns non-effectivity (hence nefness) of certain rays, which we call \emph{good rays}, in the Mori cone of the blow-up $X_n$ of the plane at $n\ge 10$ general points. Nagata's original result was the existence of a good ray for $X_n$ with $n\ge 16$ a square number. Using degenerations, we give examples of good rays for $X_n$ for all $n\ge 10$. As with Nagata's original result, this implies the existence of counterexamples to Hilbert's XIV problem. Finally we show that Nagata'…
Modifying an approach of J. Roe, this paper gives an improved lower bound on the degrees d such that for general points p1,...,pn in P2 and m > 0 there is a plane curve of degree d vanishing at each point pi with multiplicity at least m. In certain cases, for m not too large compared with n, the new bound implies a bound conjectured by Nagata.
In 2003, Shestakov-Umirbaev solved Nagata's conjecture on an automorphism of a polynomial ring. In the present paper, we reconstruct their theory by using the "generalized Shestakov-Umirbaev inequality", which was recently given by the author. As a consequence, we obtain a more precise tameness criterion for polynomial automorphisms. In particular, we show that no tame automorphism of a polynomial ring admits a reduction of type IV.
T. Szemberg proposed in 2001 a generalization to arbitrary varieties of M. Nagata's 1959 open conjecture, which claims that the Seshadri constant of r>9 very general points of the projective plane is maximal. Here we prove that Nagata's original conjecture implies Szemberg's for all smooth surfaces X with an ample divisor L generating its Neron-Severi group and such that L^2 is a square. More generally, we prove that the (n-1)-dimensional Seshadri constant of an ample divisor L on a projective variety X of dimension n at r very general points is bounded below by the product of the (n-1)-dimens…
It is well known that multi-point Seshadri constants for a small number $s$ of points in the projective plane are submaximal. It is predicted by the Nagata conjecture that their values are maximal for $s\geq 9$ points. Tackling the problem in the language of valuations one can make sense of $s$ points for any positive real $s\geq 1$. We show somewhat surprisingly that a Nagata-type conjecture should be valid for $s\geq 8+1/36$ points and we compute explicitly all Seshadri constants (expressed here as the asymptotic maximal vanishing element) for $s\leq 7+1/9$.
Prompted by results of Guardo, Van Tuyl and the second author for lines in projective 3 space, we develop asymptotic upper bounds for the least degree of a homogeneous form vanishing to order at least m on a union of disjoint r dimensional planes in projective n space for n at least 2r+1. These considerations lead to new conjectures that suggest that the well known conjecture of Nagata for points in the projective plane is not sporadic, but rather a special case of a more general phenomenon.
We give lower bounds for the degree of the discriminant with respect to y of separable polynomials f in K[x,y] over an algebraically closed field of characteristic zero. Depending on the invariants involved in the lower bound, we give a geometrical characterisation of those polynomials having minimal discriminant, and give an explicit construction of all such polynomials in many cases. In particular, we show that irreducible monic polynomials with minimal discriminant coincide with coordinate polynomials. We obtain analogous partial results for the case of nonmonic or reducible polynomials by …
Nagata solved Hilbert's 14-th problem in 1958 in the negative. The solution naturally lead him to a tantalizing conjecture that remains widely open after more than half a century of intense efforts. Using Nagata's theorem as starting point, and the conjecture, with its multiple variations, as motivation, we explore the important questions of finite generation for invariant rings, for support semigroups of multigraded algebras, and for Mori cones of divisors on blown up surfaces, and the rationality of Waldschimdt constants. Finally we suggest a connection between the Mori cone of the Zariski-R…
In connection with his counter-example to the fourteenth problem of Hilbert, Nagata formulated a conjecture concerning the postulation of r fat points of the same multiplicity in the projective plane and proved it when r is a square. Iarrobino formulated a similar conjecture in any projective space P^d. We prove Iarrobino's conjecture when r is a d-th power. As a corollary, we obtain new counter-examples modeled on those by Nagata.
Suppose that there exists a hypersurface with the Newton polytope $Δ$, which passes through a given set of subvarieties. Using tropical geometry, we associate a subset of $Δ$ to each of these subvarieties. We prove that a weighted sum of the volumes of these subsets estimates the volume of $Δ$ from below. As a particular application of our method we consider a planar algebraic curve $C$ which passes through generic points $p_1,\dots,p_n$ with prescribed multiplicities $m_1,\dots,m_n$. Suppose that the minimal lattice width $ω(Δ)$ of the Newton polygon $Δ$ of the curve $C$ is at least $\max(m_i…
A conjecture, related to the Nagata conjecture and the Segre-Harbourne-Gimigliano-Hirschowitz conjecture, states that every integral curve with negative self-intersection on the blow-up of $¶^2$ at a set of points in very general position is a (-1)-curve. In this paper we verify this conjecture for all curves whose image on $¶^2$ is a curve with a singularity of multiplicity two at one of the centers of the blow-up.
We prove that a submaximal plane curve (i.e., an irreducible counterexample to Nagata's conjecture) with r singular points has sequence of multiplicities (m, n, ..., n) with m<sn for every integer with ((s-1)(s+2))^2 > 6.76(r-1).
We consider the last value $\hatμ (ν)$ of the vanishing sequence of $H^0(L)$ along a divisorial or irrational valuation $ν$ centered at $\mathcal{O}_{\mathbb{P}^2,p}$, where $L$ resp. $p$ is a line resp. a point of the projective plane $\mathbb{P}^2$ over an algebraically closed field. This value contains, for valuations, similar information as that given by Seshadri constants for points. It is always true that $\hatμ (ν) \geq \sqrt{1 / \mathrm{vol}(ν)}$ and minimal valuations are those satisfying the equality. In this paper, we prove that the Greuel-Lossen-Shustin Conjecture implies a variati…
Let $X$ be a projective toric surface of Picard number one blown up at a general point. We bring an infinite family of examples of such $X$ whose Kleiman-Mori cone of curves is not closed: there is no negative curve generating one of the two boundary rays of the cone. These examples are related to Nagata's conjecture and rationality of Seshadri constants.
Let X_g=C^{(2)}_g be the second symmetric product of a very general curve of genus g. We reduce the problem of describing the ample cone on X_g to a problem involving the Seshadri constant of a point on X_{g-1}. Using this we recover a result of Ciliberto-Kouvidakis that reduces finding the ample cone of X_g to the Nagata conjecture when g\ge 9. We also give new bounds on the the ample cone of X_g when g=5.
Let $Z$ be a finite set of $s$ points in the projective space $\mathbb{P}^n$ over an algebraically closed field $F$. For each positive integer $m$, let $α(mZ)$ denote the smallest degree of nonzero homogeneous polynomials in $F[x_0,\ldots,x_n]$ that vanish to order at least $m$ at every point of $Z$. The Waldschmidt constant $\widehatα(Z)$ of $Z$ is defined by the limit \[ \widehatα(Z)=\lim_{m \to \infty}\frac{α(mZ)}{m}. \] Demailly conjectured that \[ \widehatα(Z)\geq\frac{α(mZ)+n-1}{m+n-1}. \] Recently, Malara, Szemberg, and Szpond established Demailly's conjecture when $Z$ is very general a…
The \emph{canonical degree} of a curve $C$ on a surface $X$ is $K_X\cdot C$. Our main result, is that on a surface of general type there are only finitely many curves with negative self--intersection and sufficiently large canonical degree. Our proof strongly relies on results by Miyaoka. We extend our result both to surfaces not of general type and to non--negative curves, and give applications, e.g. to finiteness of negative curves on a general blow--up of $\mathbb P^ 2$ at $n\geq 10$ general points (a result related to \emph{Nagata's Conjecture}). We finally discuss a conjecture by Vojta co…
Let $π: X \rightarrow \mathbb{P}^2$ be the blow-up of $\mathbb{CP}^2$ in $n$ points $x_i$ in very general position, and let $E_i$ be the exceptional divisor over $x_i$. For $0 \leq n \leq 9$ we calculate Okounkov bodies of graded linear series given by sections of multiples of line bundles $π^\ast \mathcal{O}_{\mathbb{P}^2}(d) \otimes \mathcal{O}_X(-m\sum_{i=1}^n E_i)$ with respect to a flag consisting of a line on $\mathbb{CP}^2$ and a point on the line in general position. Furthermore, we show what Nagata's Conjecture predicts on these Okounkov bodies when $n > 9$.
