We explain how logarithmic structures select principal components in an intersection of schemes. These manifest in Chow homology and can be understood using strict transforms under logarithmic blowups. Our motivation comes from Gromov–Witten theory. The toric contact cycles in the moduli space of curves parameterize curves that admit a map to a fixed toric variety with prescribed contact orders. We show that they are intersections of virtual strict transforms of double ramification cycles in blowups of the moduli space of curves. We supply a calculation scheme for the virtual strict transforms, and deduce that toric contact cycles lie in the tautological ring of the moduli space of curves. This is a higher-dimensional analogue of a result of Faber and Pandharipande. The operational Chow rings of Artin fans play a basic role, and are shown to be isomorphic to rings of piecewise polynomials on associated cone complexes. The ingredients in our analysis are Fulton’s blowup formula, Aluffi’s formulas for Segre classes of monomial schemes, piecewise polynomials, and degeneration methods. A model calculation in toric intersection theory is treated without logarithmic methods and may be read independently.

我们解释对数结构如何在方案的交集中选择主成分。这些表现在 Chow 同源性中，并且可以使用对数放大下的严格变换来理解。我们的动机来自格罗莫夫-维滕理论。曲线模空间中的复曲面接触循环参数化曲线，这些曲线允许映射到具有规定接触顺序的固定复曲面变化。我们证明它们是曲线模空间放大中双分支循环的虚拟严格变换的交点。我们提供了虚拟严格变换的计算方案，并推导出环面接触循环位于曲线模空间的同义反复环中。这是 Faber 和 Pandharipande 结果的高维模拟。 Artin 扇形的运行 Chow 环发挥着基本作用，并且与相关圆锥复合体上的分段多项式环同构。我们分析的成分是 Fulton 的爆炸公式、单项式方案 Segre 类的 Aluffi 公式、分段多项式和退化方法。复曲面相交理论中的模型计算不使用对数方法进行处理，并且可以独立读取。