There is a very recent paper by Zak : http://mathecon.cemi.rssi.ru/zak/files/Castelnuovo%20Bounds%20for%20Higher%20Dimensional%20Varieties.pdf which deals with this issue. He has found many new bounds on the total Betti numbers. For instance, he proves that if $X \subset \mathbb{P}^n$ is smooth of degree $d$ and of codimension $a$, then:

$$ b(X) \leq \dfrac{d^{\dim X +1}}{a^{\dim X}},$$

provided that the $d \geq 2(a+1)^2$ and that $n \geq 2$. He gives other bounds if the degree is lower. He also proves that these bounds are asymptotically sharp. Note the surprising fact that these bounds only depend on the degree, the dimension and the codimension.

There is a very recent paper by Zak : http://mathecon.cemi.rssi.ru/zak/files/Castelnuovo%20Bounds%20for%20Higher%20Dimensional%20Varieties.pdf which deals with this issue. He has found many new bounds on the total Betti numbers. For instance, he proves that if $X \subset \mathbb{P}^n$ is smooth of degree $d$ and of codimension $a$, then:

$$ b(X) \leq \dfrac{d^{\dim X +1}}{a^{\dim X}},$$

provided that the $d \geq 2(a+1)^2$ and that $\dim X \geq 2$. He gives other bounds if the degree is lower. He also proves that these bounds are asymptotically sharp. Note the surprising fact that these bounds only depend on the degree, the dimension and the codimension.

There is a very recent paper by Zak : http://mathecon.cemi.rssi.ru/zak/files/Castelnuovo%20Bounds%20for%20Higher%20Dimensional%20Varieties.pdf which deals with this issue. He has found many new bounds on the total Betti numbers. For instance, he proves that if $X \subset \mathbb{P}^n$ is smooth of degree $d$ and of codimension $a$, then:

$$ b(X) \leq \dfrac{d^{\dim X +1}}{a^n},$$

provided that the $d \geq 2(a+1)^2$ and that $n \geq 2$. He gives other bounds if the degree is lower. He also proves that these bounds are asymptotically sharp. Note the surprising fact that these bounds only depend on the degree, the dimension and the codimension.

There is a very recent paper by Zak : http://mathecon.cemi.rssi.ru/zak/files/Castelnuovo%20Bounds%20for%20Higher%20Dimensional%20Varieties.pdf which deals with this issue. He has found many new bounds on the total Betti numbers. For instance, he proves that if $X \subset \mathbb{P}^n$ is smooth of degree $d$ and of codimension $a$, then:

$$ b(X) \leq \dfrac{d^{\dim X +1}}{a^{\dim X}},$$

provided that the $d \geq 2(a+1)^2$ and that $n \geq 2$. He gives other bounds if the degree is lower. He also proves that these bounds are asymptotically sharp. Note the surprising fact that these bounds only depend on the degree, the dimension and the codimension.