## V.3: Monoidal Transformations 


### V.3.1.  #to-work

Let $X$ be a nonsingular projective variety of any dimension, let $Y$ be a nonsingular subvariety, and let $\pi: \tilde{X} \rightarrow X$ be obtained by blowing up $Y$. Show that $p_a(\tilde{X})=$ $p_a(X)$

### V.3.2.  #to-work

Let $C$ and $D$ be curves on a surface $X$, meeting at a point $P$. Let $\pi: \tilde{X} \rightarrow X$ be the monoidal transformation with center $P$. 

Show that 
\[
\tilde{C} \cdot \tilde{D}=C . D-\mu_P(C) \cdot \mu_P(D)
.\]
Conclude that $C . D=\sum \mu_P(C) \cdot \mu_P(D)$, where the sum is taken over all intersection points of $C$ and $D$, including infinitely near intersection points.

### V.3.3.  #to-work

Let $\pi: \tilde{X} \rightarrow X$ be a monoidal transformation, and let $D$ be a very ample divisor on $X$. Show that $2 \pi^* D-E$ is ample on $\tilde{X}$.[^hint.v.3.3]

[^hint.v.3.3]: 
Hint: Use a suitable generalization of (I, Ex. 7.5) to curves in $\mathbf{P}^n$.

### V.3.4. Multiplicity of a Local Ring.  #to-work

Let $A$ be a noetherian local ring with maximal ideal $\mfm$. For any $l>0$, let $\psi(l)=$ length $\left(A / \mfm^l\right)$. We call $\psi$ the **Hilbert-Samuel function of $A$**. 

a. Show that there is a polynomial $P_A(z) \in \mathbf{Q}[z]$ such that $P_A(l)=\psi(l)$ for all $l \gg 0$. This is the Hilbert-Samuel polynomial of $A$.[^hint.v.3.4][^v.3.4.nagata.ref]

b. Show that $\operatorname{deg} P_A=\operatorname{dim} A$.

c. Let $n=\operatorname{dim} A$. Then we define the multiplicity of $A$, denoted $\mu(A)$, to be $(n !)$. (leading coefficient of $P_A$ ). If $P$ is a point on a noetherian scheme $X$, we define the multiplicity of $P$ on $X, \mu_P(X)$, to be $\mu\left(\mathcal{O}_{P, X}\right)$.

d. Show that for a point $P$ on a curve $C$ on a surface $X$, this definition of $\mu_P(C)$ coincides with the one in the text just before (3.5.2).

e. If $Y$ is a variety of degree $d$ in $\mathbf{P}^n$, show that the vertex of the cone over $Y$ is a point of multiplicity $d$.

[^v.3.4.nagata.ref]: 
See Nagata $[7, \text{Ch} III, \S 23]$ or Zariski-Samuel $[1, \text{vol} 2 , \text{Ch} VIII, \S 10]$.

[^hint.v.3.4]: 
Hint: Consider the graded ring $\mathrm{gr}_{\mfm} A=\bigoplus_{d \geqslant 0} \mfm^d / \mfm^{d+1}$, and apply $(\mathrm{I}, 7.5)$

### V.3.5.  #to-work

Let $a_1, \ldots, a_r, r \geqslant 5$, be distinct elements of $k$, and let $C$ be the curve in $\mathbf{P}^2$ given by the (affine) equation $y^2=\prod_{i=1}^r\left(x-a_i\right)$. Show that the point $P$ at infinity on the $y$-axis is a singular point. Compute $\delta_P$ and $g(\tilde{Y})$, where $\tilde{Y}$ is the normalization of $Y$. Show in this way that one obtains hyperelliptic curves of every genus $g \geqslant 2$.

### V.3.6.  #to-work

Show that analytically isomorphic curve singularities (I, 5.6.1) are equivalent in the sense of (3.9.4), but not conversely.

### V.3.7.  #to-work

For each of the following singularities at $(0,0)$ in the plane, give an embedded resolution, compute $\delta_P$, and decide which ones are equivalent.

a. $x^3+y^5=0$.

b. $x^3+x^4+y^5=0$.

c. $x^3+y^4+y^5=0$.

d. $x^3+y^5+y^6=0$.

e. $x^3+x y^3+y^5=0$.

### V.3.8.  #to-work

Show that the following two singularities have the same multiplicity, and the same configuration of infinitely near singular points with the same multiplicities, hence the same $\delta_P$, but are not equivalent.

a. $x^4-x y^4=0$.

b. $x^4-x^2 y^3-x^2 y^5+y^8=0$.