Vakil and Matchett-Wood (Discriminants in the Grothendieck ring of varieties, 2013. arXiv:1208.3166) made several conjectures on the topology of symmetric powers of geometrically irreducible varieties based on their computations on motivic zeta functions. Two of those conjectures are about subspaces of \(\text {Sym}^n(\mathbb {P}^1)\). In this note, we disprove one of them and prove a stronger form of the other, thereby obtaining (counter)examples to the principle of Occam’s razor for Hodge structures.

中文翻译：

---

关于$$ \ text {Sym} ^ n（\ mathbb {P} ^ 1）$$ Sym n（P 1）和Occam剃刀针对Hodge结构的某些子空间的同调

Vakil和Matchett-Wood（品种在格洛腾迪克环上的判别式，2013年。arXiv：1208.3166）基于对几何不可归约品种的动力zeta函数的计算，对几何不可约品种的对称幂拓扑作了一些推测。其中两个猜想与\（\ text {Sym} ^ n（\ mathbb {P} ^ 1）\）的子空间有关。在本说明中，我们不赞成其中之一，而是证明另一种形式更强，从而获得了针对霍奇结构的Occam剃刀原理的（反例）示例。