Let $X$ be a smooth complex projective variety of dimension $n\geq1$ and let $H$ be a very ample line bundle on $X.$ Suppose $X\ncong\mathbb{P}^n.$ Fix a closed point $x\in X$ and denote by $\mathfrak{m}_x\subset\mathcal{O}_X$ its ideal sheaf. Set $V=|H\otimes\mathfrak{m}_x^2|$ to be subspace of divisors in $|H|$ which pass through $x$ with multiplicity at least $2.$ What is the dimension of the base locus of $V$?
Since $H$ is very ample, we have that $\rho\colon H^0(H)\rightarrow H^0(H/\mathfrak{m}_x^2)\cong\mathbb{C}^{n+1}$ is surjective. However this is not an isomorphism because $h^0(H)>n+1$ (otherwise $X\cong\mathbb{P}^n$), so $V=H^0(H\otimes\mathfrak{m}_x^2)=\ker(\rho)\neq\emptyset.$ If $H=2L$ for a very ample divisor $L,$ then the base locus is $\{x\}$: if $y\neq x,$ it is enough to take $H=L_1+L_2$ for $L_1,L_2\in |L|$ such that $L_1,L_2$ contains $x$ but not $y.$ However I cannot solve the general case.
Any help is greatly appreciated.
Best Answer
The base locus of $|H - 2x|$ coincides with the intersection of $X \subset \mathbb{P}^N$ and the embedded tangent space $$ \mathbb{P}^n \cong \mathrm{T}_{x}(X) \subset \mathbb{P}^N. $$ Of course, $$ \dim(X \cap \mathrm{T}_{x}(X)) \ge 2n - N, $$ and "generically" this is an equality, but there are various exceptions.