A bound on the geometric genus of projective varieties verifying certain flag conditions

Fix integers $n,r,s_{1},...,s_{l}$ and let $\mathcal {S}(n,r;s_{1},...,s_{l})$ be the set of all integral, projective and nondegenerate varieties $V$ of degree $s_{1}$ and dimension $n$ in the projective space $\mathbf {P}^{r}$, such that, for all $i=2,...,l$, $V$ does not lie on any variety of dimension $n+i-1$ and degree $<s_{i}$. We say that a variety $V$ satisfies a flag condition of type $(n,r;s_{1},...,s_{l})$ if $V$ belongs to $\mathcal {S}(n,r;s_{1},...,s_{l})$. In this paper, under the hypotheses $s_{1}>>...>>s_{l}$, we determine an upper bound $G^{h}(n,r;s_{1},...,s_{l})$, depending only on $n,r,s_{1},...,s_{l}$, for the number $G(n,r;s_{1},...,s_{l}):= {max} {\{} p_{g}(V) : V\in \mathcal {S}(n,r;s_{1},...,s_{l}){\}}$, where $p_{g}(V)$ denotes the geometric genus of $V$. In case $n=1$ and $l=2$, the study of an upper bound for the geometric genus has a quite long history and, for $n\geq 1$, $l=2$ and $s_{2}=r-n$, it has been introduced by Harris. We exhibit sharp results for particular ranges of our numerical data $n,r,s_{1},...,s_{l}$. For instance, we extend Halphen's theorem for space curves to the case of codimension two and characterize the smooth complete intersections of dimension $n$ in $\mathbf {P}^{n+3}$ as the smooth varieties of maximal geometric genus with respect to appropriate flag condition. This result applies to smooth surfaces in $\mathbf {P}^{5}$. Next we discuss how far $G^{h}(n,r;s_{1},...,s_{l})$ is from $G(n,r;s_{1},...,s_{l})$ and show a sort of lifting theorem which states that, at least in certain cases, the varieties $V\in \mathcal {S}(n,r;s_{1},...,s_{l})$ of maximal geometric genus $G(n,r;s_{1},...,s_{l})$ must in fact lie on a flag such as $V=V_{s_{1}}^{n}\subset V_{s_{2}}^{n+1}\subset ...\subset V_{s_{l}}^{n+l-1}\subset {\mathbf {P}^{r}}$, where $V^{j}_{s}$ denotes a subvariety of $\mathbf {P}^{r}$ of degree $s$ and dimension $j$. We also discuss further generalizations of flag conditions, and finally we deduce some bounds for Castelnuovo's regularity of varieties verifying flag conditions.