A Compactification of the Space of Plane Curves
| dc.creator | Hacking, Paul | |
| dc.date | 2001-04-19 | |
| dc.date.accessioned | 2026-07-07T04:41:23Z | |
| dc.date.available | 2026-07-07T04:41:23Z | |
| dc.description | We define a geometrically meaningful compactification of the moduli space of smooth plane curves, which can be calculated explicitly. The basic idea is to regard a plane curve D in P^2 as a pair (P^2,D) of a surface together with a divisor, and allow both the surface and the curve to degenerate. For plane curves of degree d at least 4, we obtain a compactification M_d which is a moduli space of stable pairs (X,D) using the log minimal model program. A stable pair (X,D) consists of a surface X such that -K_X is ample and a divisor D in a given linear system on X with specified singularities. Note that X may be non-normal, and K_X is Q-Cartier but not Cartier in general. We give a rough classification of stable pairs of arbitrary degree, a complete classification in degrees 4 and 5, and a partial classification in degree 6. The compactification is particularly simple if d is not a multiple of 3 - in particular the surface X has at most 2 components. We give a characterisation of these surfaces in terms of the singularities and the Picard numbers of the components. Moreover, we show that M_d is smooth in this case. | |
| dc.description | LaTeX, 74 pages | |
| dc.identifier | https://arxiv.org/abs/math/0104193 | |
| dc.identifier | http://arxiv.org/abs/math/0104193 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/61334 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 14H10, 14H50 | |
| dc.title | A Compactification of the Space of Plane Curves | |
| dc.type | text |