# Correction to: Irrationality issues for projective surfaces

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## 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.

*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 (

*b*,

*y*) 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.Lopez, A.F., Pirola, G.P.: On the curves through a general point of a smooth surface in \({\mathbb{P}}^3\). Math. Z.
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