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Results tagged with enumerative-geometry
Search options answers only
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user 4428
3
votes
Accepted
Counting maximally tangent conics relative to a cubic
If $X$ is a cubic and $P \in X$ is a point such that there is totally tangent at $P$ conic then
$$
6P = 2H,
$$
where $H$ is the restriction to $X$ of the line class of $\mathbb{P}^2$. Thus, the set of …
- 39.3k
6
votes
Embedding $G(2,n)$ into $G(k,n)$
Let $V = k^n$. The map in question is the composition of the canonical map
$$
f:G(2,V) \to G(k,S^{k-1}V)
$$
given by the $(k-1)$-th symmetric power of the tautological bundle, and the (noncanonical) …
- 39.3k
10
votes
Accepted
Counting curves of degree 4 in $\mathbb{P}^{3}$
Zero. Indeed, if the intersection $Q_1 \cap Q_2$ of two quadrics is singular at $p_1$, then there is a quadric $Q$ in the pencil generated by $Q_1$ and $Q_2$ which is singular at $p_1$. On the other h …
- 39.3k
2
votes
Accepted
Degree three, codimension one subvarieties lying on a quadratic hypersurface
If the linear span of $V$ has dimension $n-1$, then $V$ is a cubic hypersurface in a hyperplane. Otherwise, $V$ is a variety of minimal degree, hence it is a cone over a linear section of $\mathbb{P}^ …
- 39.3k