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<p>We consider a one-dimensional family of rational surfaces with automorphisms discussed in (Michigan Math J 56(2):315–330 2008; Indiana Univ Math J 62(4):1143–1164 2013; Algebraic Geometry and Its Applications. Aspects of Mathematics, vol. E25, pp. 39–45 1994). In a degeneration of this family, the limiting map is the identity map on a special fiber. We check that the map on the total space of the family has indeterminacy in the special fiber. However, we show that after blowing-up at an indeterminate curve, there is an induced birational map on the exceptional divisor over the indeterminate curve. Moreover, we show that this map has dynamical degree <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\lambda =16\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>=</mo> <mn>16</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

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The degeneration of a family of rational surface automorphisms

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  • 发表日期：

    2026年3月30日

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  • Qitong Jiang

摘要

We consider a one-dimensional family of rational surfaces with automorphisms discussed in (Michigan Math J 56(2):315–330 2008; Indiana Univ Math J 62(4):1143–1164 2013; Algebraic Geometry and Its Applications. Aspects of Mathematics, vol. E25, pp. 39–45 1994). In a degeneration of this family, the limiting map is the identity map on a special fiber. We check that the map on the total space of the family has indeterminacy in the special fiber. However, we show that after blowing-up at an indeterminate curve, there is an induced birational map on the exceptional divisor over the indeterminate curve. Moreover, we show that this map has dynamical degree \(\lambda =16\) λ = 16 .