Background. Let $C$ be a smooth projective curve of genus $g$. Denote by $C^d$ its d-th symmetric product and by $G^r_d(C)$ its associated variety of linear series of type $\mathfrak{g}^r_d$, i.e. $$ G^r_d(C):= \left \{ \mathcal{L} = (L, V) \: | \: L \in Pic^d(C), V \in G(r + 1, H^0(C, L)) \right \}$$
Definition. A linear series $\mathcal{L}$ on $C$ is said to admit a length $n$ de Jonquières divisor if $$ a_1 p_1 + \ldots + a_n p_n \in \mathbb{P}(V) $$ for some points $p_1,\ldots , p_n \in C$ and some positive integers $a_1, \ldots, a_n$ whose sum is $d$.
We call the corresponding divisor $$D := p_1 + \ldots + p_n \in C^n$$ a de Jonquières divisor of length $n$.
De Jonquières formula (In ACGH chapter VIII, §5) states that, if we expect there to be a finite number of de Jonquières divisors of length $n$, then this virtual number is given by the coefficient of the monomial $t_1 \cdot \ldots \cdot t_n$ in $$(1 + a_1^2 t_1 + \ldots + a_n^2 t_n)^g(1 + a_1 t_1 + \ldots + a_n t_n)^{d−r−g}.$$ .
My question.
- Why is the virtual count in this formula "wrong"? I heard there are supposed to be some counter-examples showing that the formula does not always give the "correct number", I'm curious to know: (1) what they are, and (2) what then is the formula actually counting...
- Is there a way to relate the virtual count in this formula to relative Gromov-Witten invariants (in the sense of Li-Ruan, Ionel-Parker and Jun Li) of $\mathbb{P}^r$ with respect to the curve $C$ embedded by the linear series $\mathcal{L}$ in the cases where it makes sense (i.e. lines/conics/... with prescribed tangencies. An example is the formula for the number tangential trisecant to a given curve $C \subset \mathbb{P}^3$ in ACGH, p.364)?
