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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 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 $H_1$ and $H_2$ be two planes in $\mathbb{P}^3$.Let $P$ be a set of $9$ points such that no three lie on a line. Suppose $H_1$ contains 4 of them and $H_2$ contains remaining $5$ points. Is it tr …

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 $X$ 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 $K_X$ …

Let $X$ be an irreducible hypersurface defined by a polynomial $f$ of degree $5$ in $\mathbb{P}^3$. Let the homogeneous co-ordinates is given by $[x, y, z, w]$ and let $H$ be a hyperplane given by $w= …

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

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 …

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 $ …

Suppose we are ginen $m$ points in $\mathbb{P}^3$ in general position. Can we give an effective bound on $m$ such that there is no degree $d$ irreducible hypersurface having isolated singularity along …

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 …