The Coolidge–Nagata conjecture

Abstract

Let E ⊆ 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 P 2 . In [Pal14a] the second author analyzed the log minimal model program run for the pair (X, 1 2 D), where (X, D) → (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 P 2 \ E is contracted by the process of minimalization. We prove the conjecture in the remaining cases.

Classification of planar rational cuspidal curves II. Log del Pezzo models

“…As it was shown in , in this case plays a crucial role and one can study using the modification of the logarithmic Minimal Model Program, the so‐called almost Minimal Model Program (see Section ), applying it to the pair . The guiding principle is the following conjecture, which strengthens the Coolidge–Nagata conjecture proved recently by Koras and the first author [, ]. Conjecture If is a log smooth completion of a smooth ‐acyclic surface then .…”

Classification of planar rational cuspidal curves I. C∗∗-fibrations

“…In fact not many general properties have been proved so far, mostly because the existing theory of log surfaces does not give efficient methods in case . However, recently M. Koras and the first author proved the Coolidge–Nagata conjecture [, ], which asserts that all rational cuspidal curves are Cremona equivalent to a line. The proof uses, among others, a modification of the log minimal model program (MMP) with half‐integral coefficients, as developed in , and which is based on the generalization of the notion of almost minimality (cf.…”

Dimension counts for cuspidal rational curves via semigroups

“…Given that our study focuses on the case n ≥ 3, it complements the largely topological investigations of (the existence of ) planar rational cuspidal curves carried out, e.g., in [14], [16], [25], and [27] (our bibliography is nonexhaustive; we recommend the overview given in [24]). In particular, Piontkowski conjectured in [26] that a rational plane curve of degree at least six may have at most three cusps (and checked the validity of his conjecture in degrees ≤ 20) and Koras-Palka [21] have announced that a proof of this result is forthcoming. It is natural to wonder whether a similar boundedness result holds for rational cuspidal curves in P n , n ≥ 3.…”