{"title":"艾森布-后藤猜想的曲面反例","authors":"Jong In Han, Sijong Kwak","doi":"10.1090/tran/9192","DOIUrl":null,"url":null,"abstract":"<p>It is well known that the Eisenbud-Goto regularity conjecture is true for arithmetically Cohen-Macaulay varieties, projective curves, smooth surfaces, smooth threefolds in <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper P Superscript 5\"> <mml:semantics> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"double-struck\">P</mml:mi> </mml:mrow> <mml:mn>5</mml:mn> </mml:msup> <mml:annotation encoding=\"application/x-tex\">\\mathbb {P}^5</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and toric varieties of codimension two. After J. McCullough and I. Peeva constructed counterexamples in 2018, it has been an interesting question to find the categories such that the Eisenbud-Goto conjecture holds. 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引用次数: 0
摘要
众所周知，艾森布德-戈托正则猜想对于算术上的科恩-麦考莱（Cohen-Macaulay）变体、投影曲线、光滑曲面、P 5 \mathbb {P}^5 中的光滑三褶，以及标度为二的环状变体都是成立的。在 J. McCullough 和 I. Peeva 于 2018 年构造出反例之后，如何找到艾森布德-后藤猜想成立的范畴一直是一个有趣的问题。迄今为止，尚未发现曲面反例，而已知任何维度大于或等于3的反例。在本文中，我们为 P 4 \mathbb {P}^4 中的投影面构造了艾森布德-后藤猜想的反例，并研究了投影不变式、同调性质和几何性质。反例是通过投影空间之间的二项式有理映射构造的。
Surface counterexamples to the Eisenbud-Goto conjecture
It is well known that the Eisenbud-Goto regularity conjecture is true for arithmetically Cohen-Macaulay varieties, projective curves, smooth surfaces, smooth threefolds in P5\mathbb {P}^5, and toric varieties of codimension two. After J. McCullough and I. Peeva constructed counterexamples in 2018, it has been an interesting question to find the categories such that the Eisenbud-Goto conjecture holds. So far, surface counterexamples have not been found while counterexamples of any dimension greater or equal to 3 are known.
In this paper, we construct counterexamples to the Eisenbud-Goto conjecture for projective surfaces in P4\mathbb {P}^4 and investigate projective invariants, cohomological properties, and geometric properties. The counterexamples are constructed via binomial rational maps between projective spaces.
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