Let $X$ be a smooth projective variety of dimension $n$ and $H$ an ample divisor on $X$.

I want to know the number of conditions on $D\in |mH|$ that $x$ be a point of multiplicity$=k$ on $D$. The following is the calculation.

Consider the blow-up at $x$ and the total transform of $D$ under the blow-up at $x$ contains the exceptional divisor with multiplicity$=k$.

The exceptional divisor of the blow-up at $x$ is $\mathbb{P}^{n-1}$.

I can't understand the below argument:

passing through $x$ imposes $1$ condition,

passing through $x$ with multiplicity $2$ imposes a further $n$ conditions,


passing through $x$ with multiplicity $k$ imposes a further $\binom {k+n-2}{n-1}$ conditions.

Can you explain this?


1 Answer 1


Let $z_1, \dots, z_n$ be a regular system of parameters for $X$ at $x$, so that $x$ is given by $z_1 = \dots = z_n = 0$. In order for a Cartier divisor $D$ having local equation $\sum a_{1_i, \dots i_n} z_1^{i_1} \dots z_n^{i_n}=0$ to have multiplicity $k$ at $x$ it is necessary and sufficient that all terms of total degree $<k$ are $0$. Thus, given that $D$ has multiplicity $\ge k-1$ at $x$, the additional conditions that it have degree $\ge k$ are that all of the coefficients of the degree-$(k-1)$ terms are $0$, a total of ${(k-1)+(n-1)}\choose n-1$ conditions.


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