Rational cuspidal curves on del-Pezzo surfaces

Indranil Biswas, Shane D'Mello, Ritwik Mukherjee, and Vamsi P. Pingali

Journal of Singularities
volume 17 (2018), 91-107

Received: 8 January 2017. Received in revised form: 14 March 2018.

DOI: 10.5427/jsing.2018.17f

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Abstract:

We obtain an explicit formula for the number of rational cuspidal curves of a given degree on a del-Pezzo surface that pass through an appropriate number of generic points of the surface. This enumerative problem is expressed as an Euler class computation on the moduli space of curves. A topological method is employed in computing the degenerate contribution to the Euler class.

Mathematical Subject Classification (2010):

14N35, 14J45

Author(s) information:

Indranil Biswas, School of Mathematics, Tata Institute of fundamental research, Homi Bhabha road, Mumbai 400005, India
email: indranil@math.tifr.res.in

Shane D'Mello, Department of Mathematics, Indian Institute of Science Education and Research Mohali, Knowledge city, Sector 81, Manauli PO, Sahibzada Ajit Singh Nagar, Punjab 140306
email: shane@iisermohali.ac.in

Ritwik Mukherjee, School of Mathematics, National Institute of Science Education and Research, Bhubaneswar (HBNI), Odisha 752050, India
email: ritwikm@niser.ac.in

Vamsi P. Pingali, Department of Mathematics, Indian Institute of Science, C V Raman Ave, Bengaluru, Karnataka 560012, India
email: vamsipingali@iisc.ac.in