Return to Answer

A simplest example is a curve $C$, then $\operatorname{Bir}(C)\cong\operatorname{Aut}(C)$

A careful note that $\operatorname{Bir}(X)$ is not a group scheme in general.Moreover if $X$ and $Y$ are birationl then $\operatorname{Bir}(X)$ is not isometric with $\operatorname{Bir}(Y)$

So we need to add some condition on $X$ in the sense of Minimal Model Program such that in scheme group theoretic sense $\operatorname{Bir}(X)$ good behave since we have luck of well defined multiplication rule. In fact is $X$ be a minimal model with terminal singularities it is a theorem that $\operatorname{Bir}(X)$ is a group scheme as an Abelian variety. In general $$\operatorname{Aut}^o(X)\cong\operatorname{Bir}^o(X)$$

where $\operatorname{Aut}^o$ is the identity connected component. See this paper

Let $X$ be a projective variety then we have the following isomorphism from MMP

$$\operatorname{Aut}(X_{min})\cong\operatorname{Bir}(X_{min})$$ where $X_{min}$ is the minimal model of projective variety $X$

For any surface of non-negative Kodaira dimension, we have $$\operatorname{Aut}(X)\subset\operatorname{Bir}(X)\cong\operatorname{Bir}(X_{min}) \cong\operatorname{Aut}(X_{min})$$

If $X$ be a Fano variety of $\dim\geq 5$ and Picard group is generated by anticanonical divisor of the variety $X$, then it is conjecture that $\operatorname{Aut}(X)\cong\operatorname{Bir}(X)$

Moreover for moveable log pair $(X,M_X)$ which is log Calabi-Yau pair, with at worst canonical singularities then $\operatorname{Aut}(X)\cong\operatorname{Bir}(X)$

A simplest example is a curve $C$, then $\operatorname{Bir}(C)\cong\operatorname{Aut}(C)$.

A careful note that $\operatorname{Bir}(X)$ is not a group scheme in general. Moreover if $X$ and $Y$ are birational then $\operatorname{Bir}(X)$ is not isometric with $\operatorname{Bir}(Y)$

So we need to add some condition on $X$ in the sense of Minimal Model Program such that in scheme group theoretic sense $\operatorname{Bir}(X)$ behaves well since we have lack of well defined multiplication rule. In fact if $X$ is a minimal model with terminal singularities it is a theorem that $\operatorname{Bir}(X)$ is a group scheme as an Abelian variety. In general $$\operatorname{Aut}^o(X)\cong\operatorname{Bir}^o(X)$$

where $\operatorname{Aut}^o$ is the identity connected component. See Hanamura - On the birational automorphism groups of algebraic varieties.

Let $X$ be a projective variety. Then we have the following isomorphism from MMP:

$$\operatorname{Aut}(X_\text{min})\cong\operatorname{Bir}(X_\text{min})$$ where $X_\text{min}$ is the minimal model of the projective variety $X$.

For any surface of non-negative Kodaira dimension, we have $$\operatorname{Aut}(X)\subset\operatorname{Bir}(X)\cong\operatorname{Bir}(X_\text{min}) \cong\operatorname{Aut}(X_\text{min}).$$

If $X$ is a Fano variety of $\dim\geq 5$ and its Picard group is generated by anticanonical divisor of the variety $X$, then it is conjectured (see Chel'tsov - Birationally rigid Fano varieties) that $\operatorname{Aut}(X)\cong\operatorname{Bir}(X)$.

Moreover for moveable log pair $(X,M_X)$ which is log Calabi–Yau pair, with at worst canonical singularities then $\operatorname{Aut}(X)\cong\operatorname{Bir}(X)$.

A simplest example is a curve $C$, then $Bir(C)\cong Aut(C)$

A careful note that $Bir(X)$ is not a group scheme in general.Moreover if $X$ and $Y$ are birationl then $Bir(X)$ is not isometric with $Bir(Y)$

So we need to add some condition on $X$ in the sense of Minimal Model Program such that in scheme group theoretic sense $Bir(X)$ good behave since we have luck of well defined multiplication rule. In fact is $X$ be a minimal model with terminal singularities it is a theorem that $Bir(X)$ is a group scheme as an Abelian variety. In general $$Aut^o(X)\cong Bir^o(X)$$

where $Aut^o$ is the identity connected component. See this paper

Let $X$ be a projective variety then we have the following isomorphism from MMP

$$Aut(X_{min})\cong Bir(X_{min})$$ where $X_{min}$ is the minimal model of projective variety $X$

For any surface of non-negative Kodaira dimension, we have $$Aut(X) ⊂ Bir(X) \cong Bir(X_{min}) \cong Aut(X_{min})$$

If $X$ be a Fano variety of $dim\geq 5$ and Picard group is generated by anticanonical divisor of the variety $X$, then it is conjecture that $Aut(X)\cong Bir(X)$

Moreover for moveable log pair $(X,M_X)$ which is log Calabi-Yau pair, with at worst canonical singularities then $Aut(X)\cong Bir(X)$

A simplest example is a curve $C$, then $\operatorname{Bir}(C)\cong\operatorname{Aut}(C)$

A careful note that $\operatorname{Bir}(X)$ is not a group scheme in general.Moreover if $X$ and $Y$ are birationl then $\operatorname{Bir}(X)$ is not isometric with $\operatorname{Bir}(Y)$

So we need to add some condition on $X$ in the sense of Minimal Model Program such that in scheme group theoretic sense $\operatorname{Bir}(X)$ good behave since we have luck of well defined multiplication rule. In fact is $X$ be a minimal model with terminal singularities it is a theorem that $\operatorname{Bir}(X)$ is a group scheme as an Abelian variety. In general $$\operatorname{Aut}^o(X)\cong\operatorname{Bir}^o(X)$$

where $\operatorname{Aut}^o$ is the identity connected component. See this paper

Let $X$ be a projective variety then we have the following isomorphism from MMP

$$\operatorname{Aut}(X_{min})\cong\operatorname{Bir}(X_{min})$$ where $X_{min}$ is the minimal model of projective variety $X$

For any surface of non-negative Kodaira dimension, we have $$\operatorname{Aut}(X)\subset\operatorname{Bir}(X)\cong\operatorname{Bir}(X_{min}) \cong\operatorname{Aut}(X_{min})$$

If $X$ be a Fano variety of $\dim\geq 5$ and Picard group is generated by anticanonical divisor of the variety $X$, then it is conjecture that $\operatorname{Aut}(X)\cong\operatorname{Bir}(X)$

Moreover for moveable log pair $(X,M_X)$ which is log Calabi-Yau pair, with at worst canonical singularities then $\operatorname{Aut}(X)\cong\operatorname{Bir}(X)$

A simplest example is a curve $C$, then $Bir(C)\cong Aut(C)$

A careful note that $Bir(X)$ is not a group scheme in general.Moreover if $X$ and $Y$ are birationl then $Bir(X)$ is not isometric with $Bir(Y)$

So we need to add some condition on $X$ in the sense of Minimal Model Program such that in scheme group theoretic sense $Bir(X)$ good behave since we have luck of well defined multiplication rule. In fact is $X$ be a minimal model with terminal singularities it is a theorem that $Bir(X)$ is a group scheme as an Abelian variety. In general $$Aut^o(X)\cong Bir^o(X)$$

See this paper

Let $X$ be a projective variety then we have the following isomorphism from MMP

$$Aut(X_{min})\cong Bir(X_{min})$$ where $X_{min}$ is the minimal model of projective variety $X$

For any surface of non-negative Kodaira dimension, we have $$Aut(X) ⊂ Bir(X) \cong Bir(X_{min}) \cong Aut(X_{min})$$

If $X$ be a Fano variety of $dim\geq 5$ and Picard group is generated by anticanonical divisor of the variety $X$, then it is conjecture that $Aut(X)\cong Bir(X)$

Moreover for moveable log pair $(X,M_X)$ which is log Calabi-Yau pair, with at worst canonical singularities then $Aut(X)\cong Bir(X)$

A simplest example is a curve $C$, then $Bir(C)\cong Aut(C)$

A careful note that $Bir(X)$ is not a group scheme in general.Moreover if $X$ and $Y$ are birationl then $Bir(X)$ is not isometric with $Bir(Y)$

So we need to add some condition on $X$ in the sense of Minimal Model Program such that in scheme group theoretic sense $Bir(X)$ good behave since we have luck of well defined multiplication rule. In fact is $X$ be a minimal model with terminal singularities it is a theorem that $Bir(X)$ is a group scheme as an Abelian variety. In general $$Aut^o(X)\cong Bir^o(X)$$

where $Aut^o$ is the identity connected component. See this paper

Let $X$ be a projective variety then we have the following isomorphism from MMP

$$Aut(X_{min})\cong Bir(X_{min})$$ where $X_{min}$ is the minimal model of projective variety $X$

For any surface of non-negative Kodaira dimension, we have $$Aut(X) ⊂ Bir(X) \cong Bir(X_{min}) \cong Aut(X_{min})$$

If $X$ be a Fano variety of $dim\geq 5$ and Picard group is generated by anticanonical divisor of the variety $X$, then it is conjecture that $Aut(X)\cong Bir(X)$

Moreover for moveable log pair $(X,M_X)$ which is log Calabi-Yau pair, with at worst canonical singularities then $Aut(X)\cong Bir(X)$