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Hi!

I am looking for notions of general position that are stronger than linear general position.

To illustrate, 3 points in linear general position don't lie on a line. I want a notion that would claim that 6 points in general position don't lie on a quadric.

To be more specific: Assume I have an ideal $I$ in $\mathbb{C}[x_1,\dots,x_n]$ with a set of generators $p_1,\dots,p_k$. Assume the degrees of the generators are $d_1,\dots,d_k$ respectively, I would like to claim that no $f(n,d_1,\dots,d_k)$ points in general position are the common roots of $I$.

Restrictions $I$ can be made (for example it comes for a parametrized family etc.).

Is anyone familiar with a "general position" notion that captures this intuition? Can anyone refer me to a discussion on notions of "general position" that are close or related to such a notion?

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  • $\begingroup$ If I'm understanding your question correctly (and your points lie in a projective space $\mathbb{P}^n$, then the "general position notion" you want is that ``for all $k$, the restriction map $H^0(\mathcal{O}_{\mathbb{P}^n}(k)) \to H^0(\mathcal{O}_A(k))$ is either injective or surjective''. $\endgroup$
    – Eric Larson
    Commented Apr 11, 2013 at 6:30
  • $\begingroup$ On the other hand, a conic ("circle") always passes through 4 general (in the linear sense) poins of a projective plane... $\endgroup$
    – IMeasy
    Commented Apr 11, 2013 at 6:33
  • $\begingroup$ The circle was not a good example, I'll change it. Eric, can you state your notations? ($H^0$, $\mathcal{O}$) I am not too familiar with algebraic geometric notions) $\endgroup$
    – Roi Livni
    Commented Apr 11, 2013 at 6:43
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    $\begingroup$ You could ask that all coordinates are algebraically independent. $\endgroup$
    – Yoav Kallus
    Commented Apr 11, 2013 at 13:29
  • $\begingroup$ @Roi Livni --- See Sándor Kovács's answer below for a formulation of what I said with less notation. $\endgroup$
    – Eric Larson
    Commented Apr 11, 2013 at 16:28

3 Answers 3

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The defining equation of a degree $d$ hypersurface in $\mathbb P^n$ has $n+d\choose d$ coefficients and hence these hypersurfaces may be parametrized by a projective space of dimension ${n+d\choose d} -1$. Picking a point to be contained by the hypersurface is a linear equation on the coefficients of these defining equations so any set of ${n+d\choose d} -1$ points is contained in at least one such hypersurface, but a set of $n+d\choose d$ points in general position is not.

If you take $n=2$ and $d=1$ you get that $3$ points in general position do not lie on a line, if you take $d=2$, then you get that $6$ points in general position do not lie on a quadric, etc.

As far as the definition of "in general position" goes, in this situation, one could say that a set of points is in general position if they impose independent conditions on the coefficients of the defining equations.

More generally, you can consider any parameter space of any type of subobjects of any fixed variety/scheme/space. Containing a point of this fixed space will impose a condition on the parameter space. You can define points being in general position with respect to the type of subobjects you're considering by requiring that the conditions they impose on the parameter space are independent.

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  • $\begingroup$ Thanks, That's pretty much the direction I am aiming at. 2 question: 1) Must I define general position with respect to the type of subojects or are there definitions in the literature, independent on the the type of subobjects, that work well on many interesting cases? 2) Is there a discussion in the literature that "justifies" such definitions of general position (in the sense that it captures the idea of randomly picked points) . $\endgroup$
    – Roi Livni
    Commented Apr 11, 2013 at 6:53
  • $\begingroup$ Being in general position is inherently connected to the with respect to part. I doubt that you can make a more general definition than that. What do you mean about justification? Any set of randomly picked points of the right size will be in general position according to this. $\endgroup$
    – Sándor Kovács
    Commented Apr 11, 2013 at 7:04
  • $\begingroup$ If I understand correctly, you say that a set of points are in general position, say with respect to degree $d$ hypersurfaces, if the linear equations they impose are independent. Not any linear equation would arise this way in the space of coefficients. In other words, given a probability measure over the set of points, you get a probability measure over the linear equations on the coefficient space that is supported on a small set. There is not much to say about this, but I would like to have some sort of reference (place where such a notion is defined, or at least used). $\endgroup$
    – Roi Livni
    Commented Apr 11, 2013 at 7:20
  • $\begingroup$ But when you randomly pick points, you're not putting a probability measure on the set of linear equations but on the set of points, or in other words on those linear equations that arise this way. Or, of course, you can put a probability measure on the whole space of linear equations, but then you have to pull that back via the induced map from the space of points to the space of linear equations. That map, but the way is just a $d$-uple embedding of $\mathbb P^n$. $\endgroup$
    – Sándor Kovács
    Commented Apr 11, 2013 at 16:27
  • $\begingroup$ Okay. The part that I was lacking is that, since $k$ points will define a hyperplane (in the large space of coefficients) uniquely: then the $k+1$ point can't be in a determined hyperplane. This happens with probability $1$. Thanks. $\endgroup$
    – Roi Livni
    Commented Apr 11, 2013 at 17:10
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If there is a family of objects parametrized by points $a$ in some topological space $X$ they usually say that "an object is in general position" (or "generic") to mean that $a$ belongs to an open dense subset of $X$. This notion of course depends on topology on $X$. When $X$ is an algebraic variety, say over $R$ or $C$, one can consider two natural topologies: the usual one and Zariski one. I think this applies to all examples given in the question and in the answer of Sandor.

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As I indicated in my comment, if you're looking for an intuitive and easy to state strong notion of general position, you might ask that the coordinates of the points are algebraically independent over the rationals (that is, there is no non-zero rational-coefficient polynomial in the coordinates that vanishes). This notion is used, for example in this paper on generic rigidity. This is a pretty arbitrary condition and may be unnecessarily strong, but it is also very powerful and could effectively capture the intuitive idea of a "generic" set of points.

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