Let $X/\mathbb C$ be a smooth projective variety. Suppose it is smoothly embedded in $\mathbf P^n$ as the zero locus of an ideal generated by homogeneous polynomials $f_1, f_2, \dots, f_r$ in $n+1$ variables.

Let $b_i = \dim H_i(X, \mathbf Q)$ be the $i$-th betti number of $X$. Let $b = \sum_{i=0}^{2 \dim X} b_i$.

Does anyone know an upper bound on $b$, depending only on $n$ and on the degrees of the polynomials $f_1, \dots, f_r$?

For $n=2$ and $r=1$, $X$ is a smooth plane curve of degree $d=\deg f$. Its genus is then given by $d(d-1)/2$, and we have

$$b_0 = b_2 = 1$$ $$b_1 = 2g$$

and we have $b = 2+d(d-1)$.

For general $n$ and $r$, the situation seems quite complicated. Perhaps one can work out a bound by induction, using Lefschetz pencils and the Leray spectral sequence?

I don't mind if the bound is terrible.



I have found my answer to this question in this paper of F. L. Zak which contains the bound

$$b(X) < {{N-1}\choose{\lfloor\frac{N-1}{2}\rfloor}} d^N$$

for a nonsingular variety $X \subseteq \mathbf P^N$ defined by equations of degree $\leq d$.


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