TITLE: Is this union of tangent spaces a known object in Algebraic Geometry?
QUESTION [3 upvotes]: Let $C$ be a family of smooth (all but one curve is smooth, I'll explain later) cubic curves in $\mathbb{P}_{\mathbb{C}}^2$ with the following property: given any two elements $a,b \in C$, the curves $a$ and $b$ intersect only at one point, namely $p$, with intersection multiplicity $9$. Because they only intersect once, every point in the projective plane other than $p$ lies on at most one cubic curve in $C$. Furthermore, we've shown that each point in the projective plane other than $p$ lies on exactly one cubic curve in $C$. Hence, we can assign to each point in $\mathbb{P}_{\mathbb{C}}^2$, a tangent vector. Thus we can construct a vector field on the projective plane this way. 
Concerning the one non-smooth cubic curve: It has a nodal singularity at $p$. So when we construct the vector field, we can simply take the tangent vector that corresponds with all of the other cubic curves.
I'm wondering if such a vector field that arises from a situation like this has a name, or whether it is even an interesting object to look at. 
Questions I'm wondering about: Which vector fields can be constructed in this way? Do all of the vector fields constructed in this way share certain properties? etc. I'm an undergraduate looking to write a senior thesis this year and this would be an extension of my REU project from the summer.

REPLY [1 votes]: I'm wondering if such a vector field that arises from a situation like this has a name, or whether it is even an interesting object to look at. 

Probably implicit differentiation and gradient flows, for functions on the plane. Your families might be projectivizations of $F(x,y)=c$ with $c$ variable and $F$ fixed.  Implicit differentiation gives the vector field perpendicular to yours, and gradient flow of $F$ might be the thing you have in mind, up to a choice of parameterization, or a replacement of $F$ by another function with the same level sets.
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