Using lattice theory, Hulek and Schütt proved that for every <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$m\in \mathbb {Z}_+$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>∈</mo> <msub> <mi mathvariant="double-struck">Z</mi> <mo>+</mo> </msub> </mrow> </math></EquationSource> </InlineEquation> there exists a nine-dimensional family <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {F}_m$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">F</mi> <mi>m</mi> </msub> </math></EquationSource> </InlineEquation> of K3 surfaces covering Enriques surfaces (called of base change type) having an elliptic pencil with a rational bisection of arithmetic genus m. We present a purely geometrical lattice free construction of these surfaces, that allows us to prove that generically the mentioned bisections are nodal. Moreover, we show that, for every <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$m\in \mathbb {Z}_+$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>∈</mo> <msub> <mi mathvariant="double-struck">Z</mi> <mo>+</mo> </msub> </mrow> </math></EquationSource> </InlineEquation>, the very general Enriques surface covered by a K3 surface in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {F}_m$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">F</mi> <mi>m</mi> </msub> </math></EquationSource> </InlineEquation> admits a countable set of nodal rational curves of arithmetic genus <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$(4k^2-4k+1)m+4k^2-4k$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">(</mo> <mn>4</mn> <msup> <mi>k</mi> <mn>2</mn> </msup> <mo>-</mo> <mn>4</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mi>m</mi> <mo>+</mo> <mn>4</mn> <msup> <mi>k</mi> <mn>2</mn> </msup> <mo>-</mo> <mn>4</mn> <mi>k</mi> </mrow> </math></EquationSource> </InlineEquation> for every <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$k\in \mathbb {Z}_+$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>∈</mo> <msub> <mi mathvariant="double-struck">Z</mi> <mo>+</mo> </msub> </mrow> </math></EquationSource> </InlineEquation>. As an application, we compute the linear class of the n-torsion multisection for every <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$n\in \mathbb {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation> for a general rational elliptic surface.

Nodal rational curves on Enriques surfaces of base change type

期刊论文
发表日期：

2025年11月3日

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Simone Pesatori

摘要

Using lattice theory, Hulek and Schütt proved that for every $m\in \mathbb {Z}_+$ $m \in Z_{+}$ there exists a nine-dimensional family $\mathcal {F}_m$ $F_{m}$ of K3 surfaces covering Enriques surfaces (called of base change type) having an elliptic pencil with a rational bisection of arithmetic genus m. We present a purely geometrical lattice free construction of these surfaces, that allows us to prove that generically the mentioned bisections are nodal. Moreover, we show that, for every $m\in \mathbb {Z}_+$ $m \in Z_{+}$ , the very general Enriques surface covered by a K3 surface in $\mathcal {F}_m$ $F_{m}$ admits a countable set of nodal rational curves of arithmetic genus $(4k^2-4k+1)m+4k^2-4k$ $(4 k^{2} - 4 k + 1) m + 4 k^{2} - 4 k$ for every $k\in \mathbb {Z}_+$ $k \in Z_{+}$ . As an application, we compute the linear class of the n-torsion multisection for every $n\in \mathbb {N}$ $n \in N$ for a general rational elliptic surface.