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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 a degree $6$ curve $C$ is given as $(Q_1 \cap Q_2) \cup l^2$ where $Q_i's$ are qudrics and $l$ is a line intersecting $Q_1 \cap Q_2$. Is it true that $C$ always lies on a quadric ?

What are the degree $6$ curves in a cubic surface $X$ other than complete intersection of $X$ with a quadric? Is there any such degree $6$ curve in X? If yes, then can it contain an elliptic curve o …

Suppse $X$ is an irreducible cubic surface in $\mathbb{P}^3$ singular along a line $l_1$. Then clearly there is a plane $H$ containing $l$ such that $X \cap H = l_1^2l_2$. My question is: can $X$ con …

Suppose $P$ be a zero dimensional subscheme of length $m$ in $\mathbb{P}^3$ which is in linearly general position. Further assume that there is a degree $d$ form passing through $P$. Can $P$ satisfy C …

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 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 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 $H_1$ 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 $6$ 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 $C$ 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 $Z$ 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 …

Suppose $X$ is a general smooth hypersurface of degree $\ge 6$ and $Y$ be an irreducible hypersurface of degree $\ge 2$. Let $X \cap Y$ has at least $5$ nodes. Is it possible that $4$ nodes of $X \ca …

It is known that an intersection of a general hyperpursuface of degree $\ge 5$ in $\mathbb{P}^3$ and a hyperplane can have at most $3$ nodes. My question is the following: Can these $3$ points lie o …