Abstract
Hesse claimed in [7] (and later also in [8]) that an irreducible projective hypersurface in ℙn defined by an equation with vanishing hessian determinant is necessarily a cone. Gordan and Noether proved in [6] that this is true forn≤3 and constructed counterexamples for everyn≥4. Gordan and Noether and Franchetta gave classification of hypersurfaces in ℙ4 with vanishing hessian and which are not cones, see [6, 5]. Here we translate in geometric terms Gordan and Noether approach, providing direct geometrical proofs of these results.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
C. Ciliberto, F. Russo, and A. Simis, Homaloidal hypersurfaces and hypersurfaces with vanishing Hessian, math.AG/0701596,Adv. Math. to appear.
A. Dimca and S. Papadima, Hypersurface complements, Milnor fibres and higher homotopy groups of arrangements,Ann. of Math. (2)158 (2003), 473–507.
I.V. Dolgachev, Polar Cremona transformations,Michigan Math. J. 48 (2000), 191–202.
L. Ein, Varieties with small dual variety, I,Invent. Math. 86 (1986), 63–74.
A. Franchetta, Sulle forme algebriche diS 4 aventi l'hessiana indeterminata,Rend. Mat. e Appl. (5 14 (1954), 252–257.
P. Gordan and M. Noether, Üeber die algebraischen formen, deren Hesse'sche determinante identisch verschwindet,Math. Ann. 10 (1876), 547–568.
O. Hesse, Über die Bedingung, unter welche eine homogene ganze function von nunabhängigen variabeln durch lineäre substitutionem von n andern unabhängigen variabeln auf eine homogene function sich zurückf-führen lässt, die eine variable weniger enthält,J. Reine Angew. Math. 42 (1851), 117–124.
O. Hesse, Zur theorie der ganzen homogenen functionem,J. Reine Angew. Math. 56 (1859), 263–269.
C. Lossen, When does the Hessian determinant vanish identically? (On Gordan and Noether's proof of Hesse's claim),Bull. Braz. Math. Soc. (N.S) 35 (2004), 71–82.
U. Perazzo, Sulle varietá cubiche la cui hessiana svanisce identicamente,Giornale di Matematiche (Battaglini) 38 (1900), 337–354.
R. Permutti, Su certe forme a hessiana indeterminata,Ricerche Mat. 6 (1957), 3–10.
R. Permutti, it]Su certe classi di forme a hessiana indeterminata,Ricerche Mat. 13 (1964), 97–105.
F. Russo, Tangents and secants of algebraic varieties: notes of a course, 24° Colóquio Brasileiro de Matemática,Instituto de Matemática Pura e Aplicada, Rio de Janeiro, 2003.
F.L. Zak, Determinants of projective varieties and their degrees, inAlgebraic transformation groups and algebraic varieties, 207–238, Encyclopaedia Math. Sci. 132, Springer Verlag, Berlin, 2004.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Garbagnati, A., Repetto, F. A Geometrical approach to Gordan-Noether's and Franchetta's contributions to a question posed by Hesse. Collect. Math. 60, 27–41 (2009). https://doi.org/10.1007/BF03191214
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF03191214