Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1501.05168

Quasi-translations and singular Hessians

Michiel de Bondt
View original source ↗Read source document ↗Export metadata ↓

Abstract

In 1876 in [8], the authors Paul Gordan and Max Nöther classify all homogeneous polynomials h in at most five variables for which the Hessian determinant vanishes. For that purpose, they study quasi-translations which are associated with singular Hessians. We will explain what quasi-translations are and formulate some elementary properties of them. Additionally, we classify all quasi-translations with Jacobian rank one and all so-called irreducible homogeneous quasi-translations with Jacobian rank two. The latter is an important result of [8]. Using these results, we classify all quasi-translations in dimension at most three and all homogeneous quasi-translations in dimension at most four. Furthermore, we describe the connection of quasi-translation with singular Hessians, and as an application, we will classify all polynomials in dimension two and all homogeneous polynomials in dimensions three and four whose Hessian determinant vanishes. More precisely, we will show that up to linear terms, these polynomials can be expressed in n-1 linear forms, where n is the dimension, according to an invalid theorem of Hesse. In the last section, we formulate some known results and conjectures in connection with quasi-translations and singular Hessians.

Explore related subjects

math.AG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Michiel de Bondt. 2015-01-21. Quasi-translations and singular Hessians. https://arxiv.org/abs/1501.05168

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

KEEP EXPLORING

Related papers

arXiv2026

Lawson--Deligne Classes and Applications

We construct the integral Lawson--Deligne map of weight $q=n-p-k-1$ on smooth complex projective $n$-folds using filtered currents. It lifts the Friedlander--Mazur cycle class, recovers the reduced generalized Abel--Jacobi invariant on homologically trivial classes, and is compatible with algebraic correspondences. A Picard--Fuchs separation argument applied to the conic and van Geemen normal functions on the mirror quintic determines explicit regulator subspaces modulo the full rational period group. For prescribed elliptic moduli and a suitable mirror-quintic fiber, the subspace generated by their $a$- and $b$-loop products has dimension twice the $\Q$-dimension of the period-monomial space. Moduli $i\sqrt{\ell_j}$ for distinct primes $\ell_j$ give $2^{k+1}$ independent images on varieties of dimension $p+k+2$; one repeated imaginary quadratic modulus gives dimension four for every $k\geq1$. Compatibility with known projective-bundle and blow-up decompositions yields independent exceptional subspaces on smooth rational varieties. We also compare the higher Chow composite with the Bloch--KLM regulator after lowering the Hodge filtration. The KLM representative reduces to a cut-current class, and equality with the Lawson composite is proved in degree zero and for constant-unit decomposable classes. The general positive-degree comparison is reduced to an explicit filtered-realization condition.

math.AG↗
arXiv2026

Complete quasimaps to $\mathsf{Bl}_{\mathbb{P}^s}(\mathbb{P}^r)$

We introduce a moduli space of ``complete quasimaps'' to $\mathsf{Bl}_{\mathbb{P}^s}(\mathbb{P}^r)$. The construction, following previous work for curves on projective spaces, essentially proceeds by blowing up Ciocan-Fontanine--Kim's space of quasimaps at loci where sections of line bundles are linearly dependent. We conjecture that tautological intersection numbers on these moduli spaces give enumerative counts of curves of fixed complex structure on $X$ subject to general incidence conditions, in contrast with traditional compactifications of the moduli spaces of maps. A result of Farkas guarantees that these spaces are pure of expected dimension. The conjecture is proven in dimension 2, where the main input is a Brill-Noether theorem for general curves on toric surfaces.

math.AG↗