2

I've been reading some papers lately on the topic of generic initial ideals and related stuff, and here and there the concept of a "general linear form" (or general quadric, quintic, etc.) comes up. This seems to have some special meaning in the context, but I can't figure out what it is, and I can't find any definition in any of the sources I'm reading or when I google. I know what a linear form is, but what does it mean that it's general? I'm new to the subject and self studying so I don't have anyone to ask.

For example, in M.L. Green's "Generic Initial Ideals" from 6 Lectures on Commutative algebra, say Proposition 2.7, he talks about general linear forms.

I've seen general forms talked about in reference to some Zariski open set, like in this paper, at the end of the first page, and I'm missing what that means. Can anyone clarify for me?

beep27
  • 21
  • 2
  • Typically "form" means homogenous polynomial so a linear form would be a homogenous polynomial of degree one. – CyclotomicField Apr 13 '22 at 17:32
  • Thanks for the answer - I know what a form is and what a linear form is, I'm just not sure what it means when they say a "general" linear form. It seems like it means something more than "arbitrary" in this context which is how I usually see the word general being used. – beep27 Apr 13 '22 at 17:42

1 Answers1

3

A linear form on a vector space $V$ is an element of $V^*$. Usually, when people say that something holds for a general linear form they mean that there is some Zariski-open set $U\subset V^*$ such that the statement holds for all elements of $U$. For example, a general linear form is non-zero. This extends to other contexts as soon as you have the Zariski topology on the space of your objects. As another example, it is true that a general plane cubic curve is smooth, in the sense that for a general choice of coefficients of the curve's equation the curve is smooth.

Sergey Guminov
  • 4,038
  • Okay thank you! So if we're talking about a general linear form and not any specific property holding for it, does that just mean it's a linear form that's non-zero? Or does the general form have any inherent properties in and of itself? Sorry if that question doesn't make sense, I'm just trying to understand haha – beep27 Apr 13 '22 at 17:54
  • @beep27 General basically means "almost any" in sone precise sense. Which forms are not good enough for you depends on the question. For example, a general linear form is not zero on a given vector, but the set of forms that do vanish on said vector heavily depends on the vector. – Sergey Guminov Apr 13 '22 at 18:58
  • Ah okay, thank you! I think I'm starting to understand it better now – beep27 Apr 13 '22 at 19:13