We study wall crossings in Bridgeland stability for the Hilbert scheme of elliptic quartic curves in three dimensional projective space. We provide a geometric description of each of the moduli spaces we encounter, including when the second component of this Hilbert scheme appears. Along the way, we prove that the principal component of this Hilbert scheme is a double blow up with smooth centers of a Grassmannian, exhibiting a completely different proof of this known result by Avritzer and Vainsencher. This description allows us to compute the cone of effective divisors of this component.
The Severi variety
$V_{d,n}$
of plane curves of a given degree
$d$
and exactly
$n$
nodes admits a map to the Hilbert scheme
$\mathbb{P}^{2[n]}$
of zero-dimensional subschemes of
$\mathbb{P}^{2}$
of degree
$n$
. This map assigns to every curve
$C\in V_{d,n}$
its nodes. For some
$n$
, we consider the image under this map of many known divisors of the Severi variety and its partial compactification. We compute the divisor classes of such images in
$\text{Pic}(\mathbb{P}^{2[n]})$
and provide enumerative numbers of nodal curves. We also answer directly a question of Diaz–Harris [‘Geometry of the Severi variety’, Trans. Amer. Math. Soc.309 (1988), 1–34] about whether the canonical class of the Severi variety is effective.
We study the birational geometry of Hilbert schemes of points on nonminimal 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 ½n and Y ½n , when there is a birational morphism f : X ! Y between surfaces. In this setting, N 1 ðY ½n Þ embeds in N 1 ðX ½n Þ, and we ask if the restriction of the stable base locus decomposition of N 1 ðX ½n Þ yields the respective decomposition in N 1 ðY ½n Þ i:e:, if the weak Lefschetz Principle holds. Even though the stable base loci in N 1 ðX ½n Þ fails to provide information about how the two decompositions interact, we show that the restriction of the augmented stable base loci of X ½n to Y ½n is equal to the stable base locus decomposition of Y ½n : 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.
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