This question is basically on applying the Grothendieck-Riemann-Roch theorem to finding a formula for the push-forward of a line bundle on $\mathbf{P}^r$ under a certain morphism. Since I have a lot of questions, let me begin with an example.

Let $\pi:\mathbf{P}^1 \longrightarrow \mathbf{P}^1$ be the morphism defined as $[x_0:x_1] \mapsto [x_0^n:x_1^n]$. Here $\mathbf{P}^1$ denotes the projective line over $\mathbf{C}$ and $n\geq 1$ is an integer. Note that $\pi$ is finite. For, it is locally given by the map $x\mapsto x^n$.

Let $X=\mathbf{P}^1$.

One can show that $\pi_\ast (\mathcal{O}(m))$ is given (up to isomorphism) by $$ \mathcal{O}(n(m+1)-1)\oplus \ldots \oplus \mathcal{O}(n(m+1)-1).$$ Here the sum is taken $n$ times. For example, $\pi_\ast \mathcal{O}_X \cong \mathcal{O}(n-1)^{\oplus n}$.

By the Grothendieck-Riemann-Roch theorem $$\pi_\ast \mathcal{O}(m)= \textrm{ch}^{-1}\Big(\pi_\ast\big(\textrm{ch}(\mathcal{O}(m))\cdot \textrm{td}(X)\big) \cdot \textrm{td}(X)^{-1}\Big).$$ Here we use that the Chern character $\textrm{ch}:K_0(X)\longrightarrow A^\cdot X$ is an isomorphism of rings. If $h$ denotes the class of a hyperplane $H$ of degree 1 in the Chow ring $A^\cdot X=\mathbf{Z}\oplus \textrm{Cl}(X)=\mathbf{Z}[h]/h^2$, we have that $$\textrm{ch}(\mathcal{O}(m)) = 1+mh\in \mathbf{Z}[h]/h^2.$$ We have a short exact sequence $$0\longrightarrow \mathcal{O}_X \longrightarrow \mathcal{O}(1)\oplus \mathcal{O}(1) \longrightarrow \mathcal{T}_X \longrightarrow 0.$$ Therefore, $$\textrm{td}(X) = 1+ h \in \mathbf{Z}[h]/h^2.$$ (Note that we do not need to tensor by $\mathbf{Q}$.) The formula now follows from the fact that $\pi_\ast(1+mh) =n+nmh$ and the short exact sequence $$0\longrightarrow \mathcal{O}_X(-1) \longrightarrow \mathcal{O}_X \longrightarrow \mathcal{O}_H \longrightarrow 0.$$

(There was an error in the above example. The push-forward $\pi_\ast$ is given by the multiplication by $n$ in a way.)

Now for the questions.

Q1. I would like to look at the morphism $\mathbf{P}^r\longrightarrow \mathbf{P}^r$ given by $[x_0:\ldots:x_r]\mapsto [x_0^n:\ldots:x_0^r]$. This is a finite morphism and therefore it should be possible to apply GRR in finding an expression for $\pi_\ast \mathcal{O}(m)$, right? Now, this is probably still very easy so I was wondering if there were any other results in this direction.

Q2. How would one do this without the Grothendieck-Riemann-Roch theorem? By "this", I also mean the above example for $\mathbf{P}^1$.

Q3. What is the geometric interpretation of this?

Final note. Since every vector bundle on $\mathbf{P}^1$ has a unique decomposition into twisted sheaves, we get a nice expression for $\pi_\ast \mathcal{E}$ where $\mathcal{E}$ is a vector bundle on $\mathbf{P}^1$.

This question is basically on applying the Grothendieck-Riemann-Roch theorem to finding a formula for the push-forward of a line bundle on $\mathbf{P}^r$ under a certain morphism. Since I have a lot of questions, let me begin with an example.

Let $\pi:\mathbf{P}^1 \longrightarrow \mathbf{P}^1$ be the morphism defined as $[x_0:x_1] \mapsto [x_0^n:x_1^n]$. Here $\mathbf{P}^1$ denotes the projective line over $\mathbf{C}$ and $n\geq 1$ is an integer. Note that $\pi$ is finite. For, it is locally given by the map $x\mapsto x^n$.

Let $X=\mathbf{P}^1$.

One can show that $\pi_\ast (\mathcal{O}(m))$ is given (up to isomorphism) by $$ \mathcal{O}(n(m+1)-1)\oplus \ldots \oplus \mathcal{O}(n(m+1)-1).$$ Here the sum is taken $n$ times. For example, $\pi_\ast \mathcal{O}_X \cong \mathcal{O}(n-1)^{\oplus n}$.

This is wrong. See David Speyer's response for the correct expression.

Now for the questions.

Q1. I would like to look at the morphism $\mathbf{P}^r\longrightarrow \mathbf{P}^r$ given by $[x_0:\ldots:x_r]\mapsto [x_0^n:\ldots:x_0^r]$. This is a finite morphism and therefore it should be possible to apply GRR in finding an expression for $\pi_\ast \mathcal{O}(m)$, right? Now, this is probably still very easy so I was wondering if there were any other results in this direction.

Q2. How would one do this without the Grothendieck-Riemann-Roch theorem? By "this", I also mean the above example for $\mathbf{P}^1$.

Q3. What is the geometric interpretation of this?

Final note. Since every vector bundle on $\mathbf{P}^1$ has a unique decomposition into twisted sheaves, we get a nice expression for $\pi_\ast \mathcal{E}$ where $\mathcal{E}$ is a vector bundle on $\mathbf{P}^1$.

This question is basically on applying the Grothendieck-Riemann-Roch theorem to finding a formula for the push-forward of a line bundle on $\mathbf{P}^r$ under a certain morphism. Since I have a lot of questions, let me begin with an example.

Let $\pi:\mathbf{P}^1 \longrightarrow \mathbf{P}^1$ be the morphism defined as $[x_0:x_1] \mapsto [x_0^n:x_1^n]$. Here $\mathbf{P}^1$ denotes the projective line over $\mathbf{C}$ and $n\geq 1$ is an integer. Note that $\pi$ is finite. For, it is locally given by the map $x\mapsto x^n$.

Let $X=\mathbf{P}^1$.

Now, I believe that one can show that $\pi_\ast (\mathcal{O}(m))$ is given (up to isomorphism) by $$ \mathcal{O}(n(m+1)-1).$$ Let me sketch how this goes.

By the Grothendieck-Riemann-Roch theorem $$\pi_\ast \mathcal{O}(m)= \textrm{ch}^{-1}\Big(\pi_\ast\big(\textrm{ch}(\mathcal{O}(m))\cdot \textrm{td}(X)\big) \cdot \textrm{td}(X)^{-1}\Big).$$ Here we use that the Chern character $\textrm{ch}:K_0(X)\longrightarrow A^\cdot X$ is an isomorphism of rings. If $h$ denotes the class of a hyperplane $H$ of degree 1 in the Chow ring $A^\cdot X=\mathbf{Z}\oplus \textrm{Cl}(X)=\mathbf{Z}[h]/h^2$, we have that $$\textrm{ch}(\mathcal{O}(m)) = 1+mh\in \mathbf{Z}[h]/h^2.$$ We have a short exact sequence $$0\longrightarrow \mathcal{O}_X \longrightarrow \mathcal{O}(1)\oplus \mathcal{O}(1) \longrightarrow \mathcal{T}_X \longrightarrow 0.$$ Therefore, $$\textrm{td}(X) = 1+ h \in \mathbf{Z}[h]/h^2.$$ (Note that we do not need to tensor by $\mathbf{Q}$.) The formula now follows from the fact that $\pi_\ast(h) =nh$ and the short exact sequence $$0\longrightarrow \mathcal{O}_X(-1) \longrightarrow \mathcal{O}_X \longrightarrow \mathcal{O}_H \longrightarrow 0.$$

Now for the questions.

Q1. I would like to look at the morphism $\mathbf{P}^r\longrightarrow \mathbf{P}^r$ given by $[x_0:\ldots:x_r]\mapsto [x_0^n:\ldots:x_0^r]$. This is a finite morphism and therefore it should be possible to apply GRR in finding an expression for $\pi_\ast \mathcal{O}(m)$, right? Now, this is probably still very easy so I was wondering if there were any other results in this direction.

Q2. How would one do this without the Grothendieck-Riemann-Roch theorem? By "this", I also mean the above example for $\mathbf{P}^1$.

Q3. What is the geometric interpretation of this? Can one see in an easy way that $\pi_\ast \mathcal{O}(m)$ is a line bundle?

Final note. Since every vector bundle on $\mathbf{P}^1$ has a unique decomposition into twisted sheaves, we get a nice expression for $\pi_\ast \mathcal{E}$ where $\mathcal{E}$ is a vector bundle on $\mathbf{P}^1$.

This question is basically on applying the Grothendieck-Riemann-Roch theorem to finding a formula for the push-forward of a line bundle on $\mathbf{P}^r$ under a certain morphism. Since I have a lot of questions, let me begin with an example.

Let $\pi:\mathbf{P}^1 \longrightarrow \mathbf{P}^1$ be the morphism defined as $[x_0:x_1] \mapsto [x_0^n:x_1^n]$. Here $\mathbf{P}^1$ denotes the projective line over $\mathbf{C}$ and $n\geq 1$ is an integer. Note that $\pi$ is finite. For, it is locally given by the map $x\mapsto x^n$.

Let $X=\mathbf{P}^1$.

One can show that $\pi_\ast (\mathcal{O}(m))$ is given (up to isomorphism) by $$ \mathcal{O}(n(m+1)-1)\oplus \ldots \oplus \mathcal{O}(n(m+1)-1).$$ Here the sum is taken $n$ times. For example, $\pi_\ast \mathcal{O}_X \cong \mathcal{O}(n-1)^{\oplus n}$.

By the Grothendieck-Riemann-Roch theorem $$\pi_\ast \mathcal{O}(m)= \textrm{ch}^{-1}\Big(\pi_\ast\big(\textrm{ch}(\mathcal{O}(m))\cdot \textrm{td}(X)\big) \cdot \textrm{td}(X)^{-1}\Big).$$ Here we use that the Chern character $\textrm{ch}:K_0(X)\longrightarrow A^\cdot X$ is an isomorphism of rings. If $h$ denotes the class of a hyperplane $H$ of degree 1 in the Chow ring $A^\cdot X=\mathbf{Z}\oplus \textrm{Cl}(X)=\mathbf{Z}[h]/h^2$, we have that $$\textrm{ch}(\mathcal{O}(m)) = 1+mh\in \mathbf{Z}[h]/h^2.$$ We have a short exact sequence $$0\longrightarrow \mathcal{O}_X \longrightarrow \mathcal{O}(1)\oplus \mathcal{O}(1) \longrightarrow \mathcal{T}_X \longrightarrow 0.$$ Therefore, $$\textrm{td}(X) = 1+ h \in \mathbf{Z}[h]/h^2.$$ (Note that we do not need to tensor by $\mathbf{Q}$.) The formula now follows from the fact that $\pi_\ast(1+mh) =n+nmh$ and the short exact sequence $$0\longrightarrow \mathcal{O}_X(-1) \longrightarrow \mathcal{O}_X \longrightarrow \mathcal{O}_H \longrightarrow 0.$$

(There was an error in the above example. The push-forward $\pi_\ast$ is given by the multiplication by $n$ in a way.)

Now for the questions.

Q1. I would like to look at the morphism $\mathbf{P}^r\longrightarrow \mathbf{P}^r$ given by $[x_0:\ldots:x_r]\mapsto [x_0^n:\ldots:x_0^r]$. This is a finite morphism and therefore it should be possible to apply GRR in finding an expression for $\pi_\ast \mathcal{O}(m)$, right? Now, this is probably still very easy so I was wondering if there were any other results in this direction.

Q2. How would one do this without the Grothendieck-Riemann-Roch theorem? By "this", I also mean the above example for $\mathbf{P}^1$.

Q3. What is the geometric interpretation of this?

Final note. Since every vector bundle on $\mathbf{P}^1$ has a unique decomposition into twisted sheaves, we get a nice expression for $\pi_\ast \mathcal{E}$ where $\mathcal{E}$ is a vector bundle on $\mathbf{P}^1$.