New constructions of unexpected hypersurfaces in $$\mathbb {P}^n$$ P
- PDF / 342,712 Bytes
- 18 Pages / 439.37 x 666.142 pts Page_size
- 79 Downloads / 202 Views
New constructions of unexpected hypersurfaces in Pn 3 Brian Harbourne1 · Juan Migliore2 · Halszka Tutaj-Gasinska ´
Received: 27 April 2019 / Accepted: 17 December 2019 © Universidad Complutense de Madrid 2020
Abstract In the paper we present new examples of unexpected varieties. The research on unexpected varieties started with a paper of Cook II, Harbourne, Migliore and Nagel and was continued in the paper of Harbourne, Migliore, Nagel and Teitler. Here we present three ways of producing unexpected varieties that expand on what was previously known. In the paper of Harbourne, Migliore, Nagel and Teitler, cones on varieties of codimension 2 were used to produce unexpected hypersurfaces. Here we show that cones on positive dimensional varieties of codimension 2 or more almost always give unexpected hypersurfaces. For non-cones, almost all previous work has been for unexpected hypersurfaces coming from finite sets of points. Here we construct unexpected surfaces coming from lines in P3 , and we generalize the construction using birational transformations to obtain unexpected hypersurfaces in higher dimensions. Keywords Cones · Fat flats · Special linear systems · Line arrangements · Unexpected varieties · Base loci Mathematics Subject Classification Primary 14N20; Secondary 13D02 · 14C20 · 14N05 · 05E40 · 14F05
B
Halszka Tutaj-Gasi´nska [email protected] Brian Harbourne [email protected] Juan Migliore [email protected]
1
Department of Mathematics, University of Nebraska, Lincoln, NE 68588-0130, USA
2
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA
3
Faculty of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland
123
B. Harbourne et al.
1 Introduction We work over an arbitrary algebraically closed field K except for results that depend on computer calculations, and for these we assume characteristic 0. The notion of an unexpected variety was introduced in [6]. That paper, using the results of [7], produced an example of a quartic curve in P2 which passes through a certain special set B of nine points imposing independent conditions on quartics. Thus the space of quartics passing through B has (affine) dimension 6; however, for any general point P on P2 , there exists a quartic passing through B and vanishing at P with multiplicity 3. This is unexpected since vanishing at a triple point typically imposes 6 conditions, so there should be no quartic vanishing on B and triply at P. In general, let R = K [Pn ] = K [x0 , . . . , xn ] denote the homogeneous coordinate ring of Pn ; it is a polynomial ring in n + 1 indeterminates with the standard grading in which each variable xi has degree 1. Then, given a scheme Z ⊆ Pn , we denote by I Z ⊂ R the saturated homogeneous ideal defining Z . Now let λi ⊂ Pn , 1 ≤ i ≤ r , be general linear varieties and let δi = dim λi . For integers m i ≥ 0, X = m 1 λ1 + · · · + m r λr denotes the scheme defined by the ideal I X = ∩i Iλmi i ⊆ R. We denote the Hilbert polynomial of R/I X by H X . We
Data Loading...
