Title: Transverse slices and Zariski's multiplicity conjecture for quasihomogeneous surfaces Show Chinese title

Title: 横截切片和准齐次曲面的扎里斯基重数猜想

Abstract: In this work, we study finitely determined, quasihomogeneous, corank 1 map germs $f$ from $(\mathbb{C}^2,0)$ to $(\mathbb{C}^3,0)$. We introduce the notion of the $\mu_{\boldsymbol{m},\boldsymbol{k}}$-minimal transverse slice of $f.$ Since this slice is a plane curve, we provide its (topological) normal form. In the irreducible case, under certain conditions, we prove that the number of characteristic exponents of the curve is upper semicontinuous. Furthermore, we show that every topologically trivial 1-parameter unfolding of $f=(f_1,f_2,f_3)$ (not necessarily with $\mu_{\boldsymbol{m},\boldsymbol{k}}$-minimal transverse slice) is of non-negative degree (i.e., for any additional term $\alpha$ in the deformation of $f_i$, the weighted degree of $\alpha$ is not smaller than the weighted degree of $f_i$). As a consequence, we present a proof of Zariski\' s multiplicity conjecture for families in this setting. Finally, under the $\mu_{\boldsymbol{m},\boldsymbol{k}}$-minimal transverse slice assumption, we show that Ruas\' conjecture holds. Show Chinese abstract

Abstract: 在本工作中，我们研究从$(\mathbb{C}^2,0)$到$(\mathbb{C}^3,0)$的有限确定、准齐次、余数 1 的映射芽$f$。我们引入了$\mu_{\boldsymbol{m},\boldsymbol{k}}$-最小横截面的的概念$f.$由于这个截面是一个平面曲线，我们为其提供其（拓扑）正规形式。在不可约的情况下，在某些条件下，我们证明曲线的特征指数的数量是上半连续的。 此外，我们证明了每个拓扑平凡的1参数展开式 $f=(f_1,f_2,f_3)$ (不一定具有 $\mu_{\boldsymbol{m},\boldsymbol{k}}$-最小横截面) 的次数是非负的（即，对于 $f_i$的形变中的任何额外项 $\alpha$， $\alpha$的加权次数不大于 $f_i$的加权次数）。 作为结果，我们给出了在这个情况下Zariski乘数猜想的一个证明。 最后，在 $\mu_{\boldsymbol{m},\boldsymbol{k}}$-最小横截面假设下，我们证明了Ruas猜想成立。