Bollettino dell'Unione Matematica Italiana

, Volume 11, Issue 1, pp 27–29

# Correction to: Irrationality issues for projective surfaces

Correction

## 1 Correction to: Boll. Unione Mat. Ital.  https://doi.org/10.1007/s40574-017-0116-2

The purpose of this note is to fix an imprecision in Theorem 3.3 of the original paper, which affects the correctness of the statement and leads to a mistake in Corollary 3.5 of the original paper. In particular, given a smooth surface $$S\subset {\mathbb {P}}^3$$ of degree $$d\geqslant 5$$, it is not true that all the families of curves computing the covering and the connecting gonality of S are listed in Example 3.4 of the original paper, as it is shown in the example below. We refer to the original paper for the setting and notation.

According to [1], where the correct assertion of Theorem 3.3 of the original paper is included in [1, Corollary 1.7], we introduce a notion of equivalence for families of curves covering S and admitting a $$\mathfrak {g}^1_k$$, as follows. Let $${\mathcal {C}}{\mathop {\longrightarrow }\limits ^{\pi }} T$$ be a covering family of k-gonal curves, i.e. a family of irreducible curves $$C_t=\pi ^{-1}(t)$$ endowed with a dominant morphism $$f:{\mathcal {C}}\longrightarrow S$$ such that for general $$t\in T$$, $${{\mathrm{gon}}}(C_t)=k$$ and $$f_{|C_t}$$ is birational. Following [1, Proof of Corollary 1.7], we obtain a diagramwhere $$B\longrightarrow H$$ is a finite cover of a subscheme $$H\subset {\mathrm {Hilb}}(S)$$ of the Hilbert scheme of curves on S, the family of curves $$\mathcal {E}{\longrightarrow } B$$ is the corresponding pullback of the universal family over $${\mathrm {Hilb}}(S)$$, and the restriction $$F_b:E_b\longrightarrow \{b\}\times {\mathbb {P}}^1$$ is a base-point-free $$\mathfrak {g}^1_k$$. Therefore, denoting by $$S^{(k)}$$ the k-fold symmetric product of S, we may define a morphism $$\gamma :B\times {\mathbb {P}}^1\longrightarrow S^{(k)}$$ sending a point (by) to the 0-cycle $$x_1+\dots +x_k$$ such that $$F^{-1}(b,y)=\{x_1,\ldots ,x_k\}$$. In particular, the image $$\gamma (B\times {\mathbb {P}}^1)$$ parameterizes the fibres of the k-gonal maps $$F_b:E_b\longrightarrow \{b\}\times {\mathbb {P}}^1$$ as b varies in B. Then we give the following (cf. [1, Definition 3.3]).

## Definition

Let $${\mathcal {C}}{\mathop {\longrightarrow }\limits ^{\pi }} T$$ and $${\mathcal {C}}'{\mathop {\longrightarrow }\limits ^{\pi '}} T'$$ be two covering families of k-gonal curves. We say that the families are equivalent if—up to consider open subsets of $$B\times {\mathbb {P}}^1$$ and $$B'\times {\mathbb {P}}^1$$—the images $$(\gamma (B\times {\mathbb {P}}^1))_{\mathrm {red}}$$ and $$(\gamma '(B'\times {\mathbb {P}}^1))_{\mathrm {red}}$$ coincide.

Therefore two covering families of k-gonal curves are equivalent if the two families of $$\mathfrak {g}^1_k$$ give the same 0-cycles of length k on S. Thanks to this notion, we can now state the correct assertion of Theorem 3.3 of the original paper (see [1, Corollary 1.7]).

## Theorem 3.3

Let $$S\subset {\mathbb {P}}^3$$ be a smooth surface of degree $$d\geqslant 5$$. Then the covering gonality of S is $${{\mathrm{cov.gon}}}(S)=d-2,$$ and any family of irreducible curves computing the covering gonality is equivalent to (a subfamily of) one of the families described in Example 3.4 of the original paper.

## Example

Assume that there exist two rational curves $$R_1,R_2\subset S$$. Given a birational map $$\varphi :R_1\dashrightarrow R_2$$, let $$\Sigma _\varphi$$ be the ruled surface swept out by the lines $$\ell _q$$ joining $$q\in R_1$$ and $$\varphi (q)\in R_2$$, and let $$C_\varphi :=\overline{(S\cap \Sigma _\varphi ){\smallsetminus } (R_1\cup R_2)}$$ be the curve cut out on S outside $$R_1$$ and $$R_2$$. Then $$C_\varphi$$ admits a map $$C_\varphi \dashrightarrow R_1$$ of degree $$d-2$$, which sends to $$q\in R_1$$ the points on the line $$\ell _q$$. Moreover, by considering a one dimensional family of maps $$\varphi :R_1\dashrightarrow R_2$$, we obtain a covering family of $$(d-2)$$-gonal curves which does not appear in Example 3.4 of the original paper. However, this family is equivalent to the family of curves $$D_p$$ in Example 3.4(ii) of the original paper with $$R=R_1$$ and $$p\in R_2$$, since both the families describe the same 0-cycles of length $$d-2$$ on S.

Analogously, the assertion of Corollary 3.5 of the original paper can be fixed as follows. For the sake of completeness, we include a short proof.

## Corollary 3.5

Let $$S\subset {\mathbb {P}}^3$$ be a smooth surface of degree $$d\geqslant 5$$. Then the connecting gonality of S is $${{\mathrm{conn.gon}}}(S)=d-2,$$ and any family of irreducible curves computing the connecting gonality is equivalent either to the family of tangent hyperplane sections in Example $$3.4(\mathrm{i})$$ of the original paper,  or to the family described in Example $$3.4(\mathrm{ii})$$ of the original paper.

## Proof

As observed in the discussion preceding Corollary 3.5 of the original paper, the family of tangent hyperplane sections computes the connecting gonality of S, so that $${{\mathrm{conn.gon}}}(S)=d-2$$.

Now, let $${\mathcal {C}}{\mathop {\longrightarrow }\limits ^{\pi }} T$$ be a family of $$(d-2)$$-gonal curves computing the connecting gonality of S. Hence it induces a diagram as (1), with $$\dim B\geqslant 2$$ because for general $$x,y\in S$$ there is a curve $$E_b$$ passing through them. As in [1, Proof of Corollary 1.7], the fibre $$F^{-1}(b,y)=\{x_1,\ldots ,x_{d-2}\}$$ over a general $$(b,y)\in B\times {\mathbb {P}}^1$$ consists of $$d-2$$ collinear points. Let $$\ell _{(b,y)}\subset {\mathbb {P}}^3$$ be the line containing $$F^{-1}(b,y)$$, and let $$x_{d-1},x_d\in S$$ be the points residual to $$F^{-1}(b,y)$$ in the intersection $$S\cap \ell _{(b,y)}$$.

If $$x_{d-1}$$ and $$x_d$$ coincide, then $$\ell _{(b,y)}$$ is tangent to S at $$x_d$$, so that the family $${\mathcal {C}}{\mathop {\longrightarrow }\limits ^{\pi }} T$$ is equivalent to the family of tangent hyperplane sections in Example 3.4(i) of the original paper. When instead $$x_{d-1}$$ and $$x_d$$ are distinct, we consider the ruled surface $$\Sigma _b$$ swept out by the lines $$\ell _{(b,y)}$$ with $$(b,y)\in \{b\}\times {\mathbb {P}}^1$$, and we define the curve $$Z_b:=\overline{(S\cap \Sigma _b){\smallsetminus } E_b}$$.

We claim that $$Z_b$$ consists of rational components. If one of the points $$x_{d-1},x_d\in S$$ is fixed as we vary $$(b,y)\in \{b\}\times {\mathbb {P}}^1$$, then $$Z_b$$ is described by the other point, so that $$Z_b$$ is an irreducible rational curve. If instead both $$x_{d-1}$$ and $$x_d$$ vary, we note that $$Z_b$$ is dominated by the curve $$\{(x,y)\in Z_b\times {\mathbb {P}}^{1}|x\in \ell _{(b,y)}\}$$, whose second projection gives a degree two map to $${\mathbb {P}}^1$$. Thus $$Z_b$$ admits a map $$Z_b\dashrightarrow {\mathbb {P}}^1$$ of degree two, and since Theorem 3.3 assures that $${{\mathrm{cov.gon}}}(S)\geqslant 3$$, we conclude that $$Z_b$$ is fixed as b varies on some open subset of B. Therefore, either $$Z=Z_b$$ consists of two rational components, or it is irreducible. In the latter case, as we vary $$(b,y)\in \{b\}\times {\mathbb {P}}^1$$, the 0-cycle $$x_{d-1}+x_d$$ describes a rational curve in the second symmetric product $$Z^{(2)}$$ of Z. By varying also $$b\in B$$, we obtain a two-dimensional family of rational curve in $$Z^{(2)}$$, which forces the curve Z to be rational.

Since S is not covered by rational curves, there exists a rational curve R, which is a component of $$Z_b$$ for any sufficiently general $$b\in B$$. In particular, for general $$(b,y)\in B\times {\mathbb {P}}^1$$, the line $$\ell _{(b,y)}$$ containing the fibre $$F^{-1}(b,y)$$ meets R, so that the family $${\mathcal {C}}{\mathop {\longrightarrow }\limits ^{\pi }} T$$ is equivalent to the family described in Example 3.4(ii) of the original paper. $$\square$$

## Reference

1. 1.
Lopez, A.F., Pirola, G.P.: On the curves through a general point of a smooth surface in $${\mathbb{P}}^3$$. Math. Z. 219, 93–106 (1995)