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Let $S$ be a projective surface and $L$ an ample line bundle on $S$. The Severi variety $\mathcal V_{\mathcal L,\delta}$ parametrizes curves with $\delta$ nodes and no other singularities in the linear system $\mathcal L = \mathbb P(H^0(X,L))$. In these notes the author says that "general nonsense" of deformation theory gives the lower bound $h^0(\mathcal O_S(C)) - 1 - \delta$ for the dimension of $\mathcal V_{\mathcal L,\delta}$ at a curve $C$ (at least on a surface with $h^1(\mathcal O_S) = 0$.

This is a silly question, but does this imply that if we pick $\delta$ no larger than $h^0(\mathcal O_S(C)) - 2$ then we get nodal curves with $\delta$ nodes in $\mathcal L$? Or is this a case of "if the space isn't empty, then this is its dimension"?

In the latter case, are there any general conditions that let us know the space isn't empty? I'm particularily interested in the case where $K_S$ and $L$ are positive multiples of the same ample bundle and we try to find $\delta = h^0(S,L) - c$ nodes ($c$ some hopefully small constant).

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There are del Pezzo surfaces of degree one whose singular anticanonical sections are all cuspidal: in this case, there are no nodal elements in this linear system. –  M P Jul 21 '13 at 22:40
    
@M P: Do you have a reference? Actually I'd be perfectly fine with those singularities too. I want the singularities to lower the geometric genus of curves in a given linear system, so any kinds of singularities are fine as long as we can estimate the sum $\sum d(d-1)/2$ of their multiplicities $d$. –  Gunnar Þór Magnússon Jul 22 '13 at 0:54
    
I do not have a reference, but I think that this is an example of the del Pezzo surfaces I have in mind: the wieghted sextic surface with equation $z^2+w^3=x^6+y^6$ in weighted projective space P(3,2,1,1). –  M P Jul 22 '13 at 5:51

1 Answer 1

When $S=\mathbb{P}^2$ and $\mathcal{L}= H= \mathcal{O}_{\mathbb{P}^2}(1)$, this question is discussed in Sernesi's book Deformation of Algebraic schemes, Chapter 4. In particular Corollary 4.7.19 page 266 states what follows:

(1) For every $d \geq 2$ and $0 \leq \delta \leq {d \choose 2}$ the Severi varieti $\mathcal{V}_{dH, \delta}$ is nonempty.

(2) For every $d \geq 2$ and $0 \leq \delta \leq {d-1 \choose 2}$ the Severi varieti $\mathcal{V}_{dH, \delta}$ contains irreducible curves.

These results have been partially extended on arbitrary projective surfaces. See again Sernesi's book, page 268 and the references given therein.

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Thanks a bunch! –  Gunnar Þór Magnússon Jul 22 '13 at 0:56

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