Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.39943

On the quadratic length of plane Cremona maps of degree 4

Alberto Calabri·Nguyen Thi Ngoc Giao
View original source ↗Read source document ↗Export metadata ↓

Abstract

Every non-linear plane Cremona map can be decomposed into quadratic maps, and the minimum number of quadratic maps required is called its quadratic length. It is known that plane Cremona maps of degree 3 have quadratic length either 2 or 3. In this paper, we study the quadratic length of plane Cremona maps $φ$ of degree 4. Recall that either $φ$ is de Jonquières, i.e. it has a base point of multiplicity 3, or it is not de Jonquières. In the latter case, $φ$ has quadratic length 2, whereas in the former case we prove that $φ$ has quadratic length 3, 4, or 5, and we classify those maps that reach the upper bound.

Explore related subjects

math.AG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Alberto Calabri, Nguyen Thi Ngoc Giao. 2026-09-30. On the quadratic length of plane Cremona maps of degree 4. https://arxiv.org/abs/2609.39943

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Arithmetic liftings and 2d TQFT for dormant opers of higher level

This manuscript represents an advance in the enumerative geometry of opers that takes the subject beyond our previous work. Motivated by a counting problem of linear differential equations in positive characteristic, we investigate the moduli space of opers from arithmetic and combinatorial points of view. We construct and investigate the moduli space classifying dormant $\mathrm{PGL}_n^{(N)}$-opers (i.e., dormant $\mathrm{PGL}_n$-opers of level $N$) on pointed stable curves in characteristic $p>0$. One of the key results is the generic étaleness of that space for $n=2$, which is proved by obtaining a detailed understanding of relevant deformation spaces. This fact induces a certain arithmetic lifting of each dormant $\mathrm{PGL}_2^{(N)}$-oper on a general curve to characteristic $p^N$; this lifting is called the canonical diagonal lifting. On the other hand, the generic étaleness also implies that the degree function for the moduli spaces in the rank $2$ case satisfies factorization properties determined by various gluing morphisms of the underlying curves. That is to say, the degree function forms a $2$d TQFT (= a $2$-dimensional topological quantum field theory); it leads us to describe dormant $\mathrm{PGL}_2^{(N)}$-opers in terms of edge numberings on trivalent graphs, as well as lattice points inside generalized rational polytopes. These results yield an effective way of computing the numbers of such objects and $2$nd order differential equations in characteristic $p^N$ with a full set of solutions.

math.AG↗
arXiv2026

A new approach to flatness in categories of sheaves

Recently, many authors have embraced the study of certain properties of modules such as projectivity, injectivity and flatness from an alternative point of view. This way, Durgun has introduced absolutely pure domains of modules as a mean to gauge the extent of the flatness of any given module (so not just to determine whether or not the module is flat). In this paper, we develop a new treatment of the goemetric flatness in categories of sheaves which shed more light on some of its various important aspects. Namely, in terms of this new approach, some classical schemes are characterized, some new results are established and also known ones are generalized.

math.AG↗
arXiv2026

Unirationality is the Same as Rational Connectedness in Characteristic Zero

In this paper we describe a fibration for a smooth, projective variety $ X $ over a field of characteristic zero. This fibration is similar to the MRC fibration, and we call it the MU fibration of $ X $. The MU fibration $ π: X \dashrightarrow MU(X) $ is characterized by the following properties: i) The very general fibres of $ π$ are unirational, ii) if $ Z $ is a unirational sub-variety of $ X $, $ z $ is a very general point of $ MU(X) $ (i.e., a point in the complement of a countable union of Zariski closed sub-sets of $ MU(X) $), and $ Z $ intersects $ π^{-1}(z) $ non-trivially, then $ Z $ is contained in $ π^{-1}(z) $, iii) The variety $ MU(X) $ is unique up to birational equivalence. If we call $ MU(X) $ a maximal unirational quotient, then $ X $ is unirational if and only if the dimension of any maximal unirational quotient is equal to zero. We use this work to show that unirationality, rational connectedness, and rational chain connectedness are equivalent for smooth varieties over a field of characteristic zero, and that the MRC quotient of a smooth, projective variety over a field of characteristic zero is not uniruled.

math.AG↗