This comes as a question in Beauville's surfaces book (Chap2.20.(2)).

Let $C$ be an irreducible curve on a surface $S$, $p\in C$ and $C^{'}$ the strict transform of $C$ on the blow-up with center $p$.

The question is:

1. $\sum m_x(C^')\le\sum m_x(C^'\cap E)$, where the sum runs through $x \in C^'$ lying over $p$ ($E$: exceptional curve).

2. find an example with strict inequality.

I can't do them. Also I don't know I understand them.

In case that $C$ :cusp and $p\in C$:singular point , can it be answer of 2)?

Any idea or help for 1) & 2)?

This comes as a question in Beauville's surfaces book (Chap2.20.(2)).

Let $C$ be an irreducible curve on a surface $S$, $p\in C$ and $C'$ the strict transform of $C$ on the blow-up with center $p$.

The question is:

1. $\sum m_x(C')\le\sum m_x(C' \cap E)$, where the sum runs through $x \in C'$ lying over $p$ ($E$: exceptional curve).

2. find an example with strict inequality.

I can't do them. Also I don't know I understand them.

In case that $C$ :cusp and $p\in C$:singular point , can it be answer of 2)?

Any idea or help for 1) & 2)?