On the birational geometry of Hilbert schemes of points and Severi divisors

Abstract

We study the birational geometry of Hilbert schemes of points on non-minimal surfaces. In particular, we study the weak Lefschetz Principle in the context of birational geometry. We focus on the interaction of the stable base locus decomposition (SBLD) of the cones of effective divisors of $X^{[n]}$ and $Y^{[n]},$ when there is a birational morphism $f:X\to Y$ between surfaces. In this setting, $N^1(Y^{[n]})$ embeds in $N^1(X^{[n]}),$ and we ask if the restriction of the stable base locus decomposition of $N^1(X^{[n]})$ yields the respective decomposition in $N^1(Y^{[n]})$ $i.e.,$ if the weak Lefschetz Principle holds. Even though the stable base loci in $N^1(X^{[n]})$ fails to provide information about how the two decompositions interact, we show that the restriction of the augmented stable base loci of $X^{[n]}$ to $Y^{[n]}$ is equal to the stable base locus decomposition of $Y^{[n]}.$ We also exhibit effective divisors induced by Severi varieties. We compute the classes of such divisors and observe that in the case that X is the projective plane, these divisors yield walls of the SBLD for some cases.

Keywords:

• Hilbert scheme
• movable cone
• Severi divisor
• stable base locus decomposition

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 14E30 (Primary)
• 14E05
• 14D22 (Secondary)

Acknowledgments

We would like to thank Gabriel Bujokas, Izzet Coskun, Joe Harris, Rob Lazarsfeld, Cristian Martínez, Alex Massarenti, Benjamin Schmidt, and Edoardo Sernesi for useful conversations about this project. Special thanks to Edoardo Sernesi for sharing with us his insights in the proof of Theorem 4.7. The first author is also grateful to the Dipartimento di Matematica e Fisica in Roma Tre for providing ideal working conditions where part of this work was done. We also thank the organizers of the conference II CNGA at CMO where parts of this work were carried out.

Additional information

Funding

During the preparation of this article the first author was partly supported by the CONACYT grant CB-2015/253061, and the second author was partially supported by the NSF grant DMS-1547145.