[Invariants des singularities des variétés sécantes de courbes]
Consider a smooth projective curve and a given embedding into projective space via a sufficiently positive line bundle. We can form the secant variety of $k$-planes through the curve. These are singular varieties, with each secant variety being singular along the previous one. We study invariants of the singularities for these varieties. In the case of an arbitrary curve, we compute the intersection cohomology in terms of the cohomology of the curve. We then turn our attention to rational normal curves of even degree. In this setting, we prove that all of the secant varieties are rational homology manifolds, meaning their singular cohomology satisfies Poincaré duality. We then compute the nearby and vanishing cycles for the largest nontrivial secant variety, which is a projective hypersurface.
Étant donnés une courbe projective lisse et un plongement donné dans l’espace projectif via un fibré en droites suffisamment positif, nous pouvons former la variété sécante des $k$-plans passant par la courbe. Ce sont des variétés singulières, chaque variété sécante étant singulière le long de la précédente. Nous étudions les invariants des singularités de ces variétés. Dans le cas d’une courbe arbitraire, nous calculons la cohomologie d’intersection en termes de cohomologie de la courbe. Nous nous intéressons ensuite aux courbes rationnelles normales de degré pair. Dans ce cadre, nous prouvons que toutes les variétés sécantes sont des variétés d’homologie rationnelle, ce qui signifie que leur cohomologie singulière satisfait à la dualité de Poincaré. Nous calculons ensuite les cycles proches et les cycles évanescents pour la plus grande variété sécante non triviale, qui est une hypersurface projective.
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Keywords: Perverse sheaves, Hodge theory, nearby and vanishing cycles, secant varieties
Mots-clés : Faisceaux pervers, théorie de Hodge, cycles proches et évanescents, variété des sécantes
Daniel Brogan 1
CC-BY 4.0
@article{JEP_2026__13__1_0,
author = {Daniel Brogan},
title = {Invariants of the singularities of secant~varieties of curves},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {1--39},
year = {2026},
publisher = {\'Ecole polytechnique},
volume = {13},
doi = {10.5802/jep.321},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.321/}
}
TY - JOUR AU - Daniel Brogan TI - Invariants of the singularities of secant varieties of curves JO - Journal de l’École polytechnique — Mathématiques PY - 2026 SP - 1 EP - 39 VL - 13 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.321/ DO - 10.5802/jep.321 LA - en ID - JEP_2026__13__1_0 ER -
%0 Journal Article %A Daniel Brogan %T Invariants of the singularities of secant varieties of curves %J Journal de l’École polytechnique — Mathématiques %D 2026 %P 1-39 %V 13 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.321/ %R 10.5802/jep.321 %G en %F JEP_2026__13__1_0
Daniel Brogan. Invariants of the singularities of secant varieties of curves. Journal de l’École polytechnique — Mathématiques, Tome 13 (2026), pp. 1-39. doi: 10.5802/jep.321
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