Smoothable locally non Cohen–Macaulay multiple structures on curves

Abstract

In this article we show that a wide range of multiple structures on curves arise whenever a family of embeddings degenerates to a morphism $\varphi$ of degree $n$. One could expect to see, when an embedding degenerates to such a morphism, the appearance of a locally Cohen–Macaulay multiple structure of certain kind (a so-called rope of multiplicity $n$). We show that this expectation is naive and that locally non Cohen–Macaulay multiple structures also occur in this situation. In seeing this we find out that many multiple structures can be smoothed. When we specialize to the case of double structures we are able to say much more. In particular, we find numerical conditions, in terms of the degree and the arithmetic genus, for the existence of many locally Cohen–Macaulay and non Cohen–Macaulay smoothable double structures. Also, we show that the existence of these double structures is determined, although not uniquely, by the elements of certain space of vector bundle homomorphisms, which are related to the first order infinitesimal deformations of $\varphi$. In many instances, we show that, in order to determine a double structure uniquely, looking merely at a first order deformation of $\varphi$ is not enough; one needs to choose also a formal deformation.

Appendix

We take advantage of this opportunity to fix a gap in the article [5], written by the first and third author. The gap concerns the arguments used there to prove [5, Theorem 3.5]. [5, Theorem 3.5] says that $K3$ carpets supported on rational normal scrolls can be smoothed. [5, Theorem 3.5] is nevertheless true and a different, independent proof of it was given in [4, Corollary 2.9].

We explain first what the problem is with the argument in [5] and then we outline the way to fix it. Precisely, the problem lies in [5, Lemma 3.2], which is false as stated. The main thesis of [5, Lemma 3.2] is this:

Let $\fancyscript{X}$ be a flat family of irreducible varieties over a smooth irreducible algebraic curve $T$ which is mapped to relative projective space by a morphism $\Phi$ induced by a relatively complete linear series. Assume that $\Phi _t$ is an embedding for all $t \ne 0$ ($0 \in T$) and $\Phi _0$ is a finite morphism of degree $2$. Let $H$ be a hyperplane in projective space. Then $\Phi (\fancyscript{C}) \cap (H \times T)$ is flat.

This claim is false in general. Indeed, if it were true, using [5, Theorem 2.1] and arguing like in [5, Corollary 3.3] we would show that the double structure $\widetilde{Y}$ that appears in Corollary 2.1 is always a ribbon and this is false as remarked in Observation 1.6. The mistake in the proof of [5, Lemma 3.2] is that, when we tensor the exact sequence

$$\begin{aligned} 0 \longrightarrow \fancyscript{O}_{\fancyscript{Y}} \overset{\alpha }{\longrightarrow }\Phi _*\fancyscript{O}_{\fancyscript{C}} \longrightarrow \fancyscript{F} \longrightarrow 0 \end{aligned}$$

with $\fancyscript{O}_{\mathbf P^n_T}/\fancyscript{I}(H \times T)$, the resulting sequence does not necessarily remain exact on the left.

We point out now how to avoid using [5, Lemma 3.2] when proving [5, Theorem 3.5]. The proof of [5, Theorem 3.5] is based on [5, Proposition 3.4], which says that the flat limit at $t=0$ of a family of $K3$ surfaces $\fancyscript{Y}_t$ embedded by a very ample polarization $\zeta _t$ is a $K3$ carpet provided that $(\fancyscript{Y}_0, \zeta _0)$ is a hyperelliptic polarized $K3$ surface. [5, Proposition 3.4] was proved using [5, Theorem 2.1] and [5, Theorem 2.1] essentially says the following:

A double structure $D$ of dimension $m$, supported on $D_{red}$ is locally Cohen–Macaulay if and only if through every closed point of $D$ there exists locally a Cartier divisor $h$ that cuts out on $D$ a locally Cohen–Macaulay double structure of dimension $m-1$, supported on the restriction of $h$ to $D_{red}$.

It was in the process of applying [5, Theorem 2.1] to prove [5, Proposition 3.4] that we used [5, Lemma 3.2]. Thus we outline now a different argument avoiding the use of [5, Lemma 3.2]. If the flat family $(\fancyscript{C},\zeta )$ over $T$ and the relative hyperplane section $H \times T$ of [5, Proposition 3.4] are suitably chosen, then [5, Theorem 2.1] can be applied to $\fancyscript{Y}_0$ in the same way as in the proof of [5, Proposition 3.4]. Precisely, let $\tilde{\varphi }$ be the restriction of $\Phi _\zeta$ to $H \times \Delta$. If one is able to choose $(\fancyscript{C},\zeta )$ and the relative hyperplane section $H \times T$ so that $\tilde{\varphi }$ corresponds to a surjective homomorphism in Hom$(\fancyscript{I}/\fancyscript{I}^2, \fancyscript{E})$, then $\fancyscript{Y}_0 \cap H$ will be a canonical ribbon. Then [5, Proposition 3.4] would follow from [5, Theorem 2.1].