arXiv subjects

J. Roé

Publications and source records attributed to J. Roé.

4 recordsLinked to original sources

arXiv2019

Unexpected surfaces singular on lines in $\mathbb{P}^3$

We study linear systems of surfaces in $\mathbb{P}^3$ singular along general lines. Our purpose is to identify and classify special systems of such surfaces, i.e., those nonempty systems where the conditions imposed by the multiple lines are not independent. We prove the existence of four surfaces arising a(projective) linear systems with a single reduced member, which numerical experiments had suggested must exist. These are unexpected surfaces and we expect that our list is complete, i.e. it contains all special linear systems of affine dimension $1$, whose projectivisation has one, reduced and irreducible member. As an application we find upper bounds for Waldschmidt constants along certain sets of general lines.

math.AG↗

arXiv2016

Newton-Okounkov bodies sprouting on the valuative tree

Given a smooth projective algebraic surface X, a point O in X and a big divisor D on X, we consider the set of all Newton-Okounkov bodies of D with respect to valuations of the field of rational functions of X centred at O, or, equivalently, with respect to a flag (E,p) which is infinitely near to O, in the sense that there is a sequence of blowups mapping the smooth, irreducible rational curve E to O. The main objective of this paper is to start a systematic study of the variation of these infinitesimal Newton-Okounkov bodies as (E, p) varies, focusing on the case X = P2.

math.AG↗

arXiv2012

Variations on Nagata's Conjecture

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's conjecture for $n\le 89$ combined with a stronger conjecture for $n=10$ implies Nagata's conjecture for $n\ge 90$.

math.AG↗

arXiv2008

An inequality between multipoint Seshadri constants

Let X be a projective variety of dimension n and L be a nef divisor on X. Denote by e_d(r;X,L) the d-dimensional Seshadri constant of r very general points in X. We prove that e_d(rs;X,L) >= e_d(r;X,L)e_d(s;P^n,O_{P^n}(1)).

math.AG↗