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Degenerations of Pascal lines

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Abstract

Let \({\mathcal {K}}\) denote a nonsingular conic in the projective plane. Pascal’s theorem says that, given six distinct points A, B, C, D, E, F on \({\mathcal {K}}\), the three intersection points \(AE \cap BF, AD \cap CF, BD \cap CE\) are collinear. The line containing them is called the Pascal line of the sextuple. However, this construction may fail when some of the six points come together. In this paper, we find the indeterminacy locus where the Pascal line is not well-defined and then use blow-ups along polydiagonals to define it. We analyse the geometry of Pascals in these degenerate cases. Finally we offer some remarks about the indeterminacy of other geometric elements in Pascal’s hexagrammum mysticum.

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Notes

  1. The rationale behind this term is explained in Chipalkatti (2018).

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Acknowledgements

The second author was supported in part by a PIMS postdoctoral fellowship and an NSERC postdoctoral fellowship.

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Authors and Affiliations

  1. Department of Mathematics, University of Manitoba, Winnipeg, MB, R3T 3R5, Canada

    Jaydeep Chipalkatti

  2. Department of Mathematics, McMaster University, Hamilton, ON, L8S 4L8, Canada

    Sergio Da Silva

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  1. Jaydeep Chipalkatti
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  2. Sergio Da Silva
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Correspondence to Sergio Da Silva.

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Chipalkatti, J., Da Silva, S. Degenerations of Pascal lines. Beitr Algebra Geom 64, 761–781 (2023). https://doi.org/10.1007/s13366-022-00655-x

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  • DOI: https://doi.org/10.1007/s13366-022-00655-x

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