1
vote
0answers
40 views

Does this proof that the resultant provides an upper bound for intersection multiplicity look correct?

Let $f,g \in \mathbb{C}[x,y]$ such that $f(0,0)=g(0,0)=0$ and the varieties $V(f)$ and $V(g)$ are both smooth at $(0,0)$ such that the tangent line of $V(f)$ and $V(g)$ is not the $y$ axis. Let ...
3
votes
1answer
109 views

Intersections of tangents with cubic are colinear

I am trying to do Exercise 5.33 on page 64 of Fulton's book on algebraic curves. 5.33 Let $C$ be an irreducible cubic, $L$ a line such that $L\bullet C = P_1+ P_2 + P_3,$ $P_i$ distinct. Let $L_i$ ...
3
votes
0answers
99 views

When is an intersection of varieties finite

Consider the general Bezout's theorem: If $p_1 \ldots p_n$ are polynomials with degrees $d_1,\ldots, d_n$ in $\mathbb{R}[x_1,\ldots,x_n]$, with $V = \{a=(a_1,\ldots, a_n) | p_i(a) = 0, \forall i\}$ ...
2
votes
2answers
333 views

When to read of the degree of a variety from its defining polynomials

The question concerns algebraic varieties. I just read the question The degree of an algebraic curve in higher dimensions and great answer by user M P. One of the thing he says is that if a curve in ...
4
votes
1answer
130 views

Very special rational points on curves over number fields

For some reason, I'm convinced the answer to the following question should be (obviously) negative, but I can't come up with a good reason. Does there exist a number field $K$, a smooth projective ...
2
votes
1answer
125 views

self-intersections in a product of two curves

Let $X$ be a smooth projective curve of genus $g$ over an algebraically closed field and consider the intersection pairing on the surface $X \times X$. I remember hearing that $\Delta^2 = 2-2g$: how ...