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Let $Y$ be a intersection of two smooth hypersurfaces of degree $m$ and $n$ in $\mathbb{P}^3$, where $m, n \ge 5$. Then my question is how many node $Y$ can have ?

Let $X$ be a very general smooth hypersurface of degree $d (\ge 5)$ in $\mathbb{P}^3$ and $Y$ be another smooth hypersurface of degree $d^{\prime}$, where $3 \le d^{\prime} \le (d-1)$ such that $X \ca …

Let $C$ be a smooth projective curve in a surface $S$. Suppose $E$ is a vector bundle of rank $r$ on $C$. Then what is the total Chern class of the sheaf $i_*E$, where $i$ is an embedding of $C$ in $S …

Let $X$ be a very general surface of degree $\ge 5$ in $\mathbb{P}^3$ and $ Y$ is arbitrary irreducible cubic hypersurface. Is $X \cap Y$ reduced ?

Let $X$ be a very general hypersurface of degree $\ge 5$ and $Y$ be an irreducible cubic hypersurface in $\mathbb{P}^3$. It is known that $X \cap H$, where $H$ is a hyperplane, can have at most $3$ no …

Let $H$ be a hypeplane in $\mathbb{P}^3$ containing a point $p$ and $I_p$ be the ideal sheaf corresponding to $p$. Consider the natural exact sequence : $0 \to \mathcal{O} \to \mathcal{O}(H) \to \ma …

Let $X$ be a finite set of points in $\mathbb{P}^3$ of cardinality $\ge 3d +3$ which fails to impose independent conditions on sections of $\mathcal{O}(d)$ and $X$ does not pass through any quadratic …

Alexander-Hirschowitz Theorem: Fix $r \ge 2$ and $d \ge 2$, and consider the linear system $L = L^{(r )}_d(-\sum_{i=1}^n 2p_i)$ consisting of hypersurfaces of degree at most $d$ in $r$ variables t …

Let $Z$ be a zero dimensional subscheme in $\mathbb{P}^3$ and $H$ be a hyperplane. Then we have an exact sequence $0 \to I_Z \to I_Z(1) \to^f I_Z(1) \mid_H \to 0$. On the other hand, we another exact …

Let $X$ be a very general hypersurface of degree $6$ in $\mathbb{P}^3$. Fix an integer $d$. Define $Y:= \{ C \in \mathbb{P}(H^0(\mathcal{O}(3))) \text{ such that } \text{dim}(\text{ Hilb}^d(X \cap C)) …

Let $X$ be a zero dimensional subscheme in $\mathbb{P}^3$ and $H$ be a hyperplane. Let $X^\prime$ be the residual subscheme with respect to $H$. Then there is an exact sequence of the form, $0 \to \ …

Let $X$ be the double cover of $\mathbb{P}^2$ branched along a divisor which is union of two lines. Then what will be the $\text{Pic}(X)$ ? Is it torsion free ? If yes, then what is its generator ?

Let $X$ be a smooth quasi-projective variety and $Y$ be a positive dimensional subvariety. Let $Z$ be a variety obtained from $X$ by contracting $Y$. My question is what is the relation between $K_X$ …

Is it true that the singular locus of an irreducible hypersurface in $\mathbb{P}^3$ have pure co-dimension ?

Is it true that one has an exact sequence of the following form: $$0 \to \mathcal{O}_Z \to I_{Z, \mathbb{P}^3}(1)\otimes \mathcal{O}_H \to I_{Z, H}(1) \to 0,$$ where $Z$ is a finite set of points in $ …