Moduli of double covers and degree one del Pezzo surfaces
- PDF / 310,841 Bytes
- 13 Pages / 439.37 x 666.142 pts Page_size
- 2 Downloads / 190 Views
Moduli of double covers and degree one del Pezzo surfaces Kenneth Ascher1 · Dori Bejleri2
Received: 24 February 2020 / Revised: 27 July 2020 / Accepted: 15 August 2020 © Springer Nature Switzerland AG 2020
Abstract Given a degree one del Pezzo surface with canonical singularities, the linear series generated by twice the anti-canonical divisor exhibits the surface as the double cover of the quadric cone branched along a sextic curve. It is natural to ask if this description extends to the boundary of a compactification of the moduli space of degree one del Pezzo surfaces. The goal of this paper is to show that this is indeed the case. In particular, we give an explicit classification of the boundary of the moduli space of anti-canonically polarized broken del Pezzo surfaces of degree one as double covers of degenerations of the quadric cone. Keywords Moduli spaces of higher dimensional varieties · Del Pezzo surfaces · Elliptic surfaces Mathematics Subject Classification 14J10 · 14J27
1 Introduction The anti-canonical linear series of del Pezzo surfaces have a rich geometric structure. For degree one del Pezzo surfaces, the linear series |− K X | is a pencil of elliptic curves with a unique base point, and the blowup of this base point gives a rational
This work was partially completed while the authors were in residence at MSRI in Spring 2019 (NSF No. DMS-1440140). Both authors supported in part by NSF Postdoctoral Fellowships. K.A. supported in part by funds from NSF Grant DMS-2001408.
B
Kenneth Ascher [email protected] Dori Bejleri [email protected]
1
Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544-1000, USA
2
Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 02138, USA
123
K. Ascher, D. Bejleri
elliptic surface with a section. On the other hand, the linear system |− 2K X | exhibits the surface X as a double cover of P(1, 1, 2), i.e., the quadric cone, branched along a sextic curve. At the same time, understanding modular compactifications of the space of degree one del Pezzo surfaces is a problem with a long history (see e.g. [2,10,11,16– 18]). In [5], we construct and describe an explicit modular compactification R of this space using the theory of stable pairs. The boundary of this space parametrizes anticanonically polarized broken del Pezzo surfaces of degree one — slc surfaces X such that K X is anti-ample and K X2 = 1 (see [5, Theorem 1.1]). In light of this, it is natural to ask whether the description of a degree one del Pezzo surface as a double cover of P(1, 1, 2) extends along the boundary of this moduli space. That is, are anti-canonically polarized broken del Pezzo surfaces of degree one also double covers of some degenerations of P(1, 1, 2)? The main goal of this paper is to show that this is indeed the case. Let D1,s ⊂ R denote the moduli stack of degree one del Pezzo surfaces with canonical singularities. If Q3 denotes the stack of pairs (Q, C), where Q is a quadric cone in P3 and C ⊂ Q is a co
Data Loading...
