Let $C$ be be a connected reducible nodal curve over alg closed field $k$, such that all (finitely many) irred components $C_i$ of $C$ are smooth and intersections between different components are nodal singularities. (Mainly I'm interested in case $C$ is reducible stable curve.

Question: What are the standard techniques to calculate the genus $g_C:= H^1(C, \mathcal{O}_C)$?

Let denote $C= C_1 \cup C_2 \cup ... \cup C_n $. One technique which is familar to me is to normalize $f: \tilde{C} \to C$ and analyze the long cohomology sequence associated to exact seq

$$ 0 \to \mathcal{O}_C \to f_*\mathcal{O}_{\tilde{C}} \to \mathcal{F} \to 0 $$

Then by definition $\tilde{C}= \dot\bigcup_i ^n C_i $, so $g_{\tilde{C}}= \sum_{i=1}^n H^1(C_i, \mathcal{O}_{C_i}) $ (because after normalization the components become disjoint and all compt's were assumed to be smooth) and $\mathcal{F}$ has $0$-dimensional support, using this we get information about $g_C$. This should work.

But in this question I would like to find out if it's also possible to use inductive arguments (a la Mayer-Vietoris techniques) which appears more "natural" to me from geometrical point of view:

We can decompose $C$ as $D \cup C_n$ for $D:=C_1 \cup ... \cup C_{n-1}$. Let $\mathcal{I}(C_n), \mathcal{I}(D) \subset \mathcal{O}_C$ be the vanishing ideals wrt $C_n$ resp $D$ considered as subschemes on $C$. Then we obtain two natural sequences

$$ 0 \to \mathcal{I}(C_n) \to \mathcal{O}_C \to (i_{C_n})_*\mathcal{O}_{C_n} \to 0 $$

and

$$ 0 \to \mathcal{I}(D) \to \mathcal{O}_C \to (i_{D})_*\mathcal{O}_{D} \to 0 $$

Can we say something interesting about the cohomology of $\mathcal{I}(C_n)$ and $\mathcal{I}(D)$ cutting out whole components?

Let consider the most simple case I was able to think of: Say $C:= V_+(X \cdot Y) \subset \mathbb{P}^2= \operatorname{Proj}(k[X,Y,Z])$ is the "cross" with $C= V_+(X) \cup V_+(Y)$. In terms of above we have the sequence

$$ 0 \to \mathcal{I}(V_+(\overline{X})) \to \mathcal{O}_C \to (i_{V_+(X)})_*\mathcal{O}_{V_+(X)} \to 0 $$

where we consider $\overline{X}$ as homogeneous element in $ \in k[X,Y,Z]/(XY)$. Do we know something about cohomology of $\mathcal{I}(V_+(\overline{X})) $ regarded as ideal sheaf in $\mathcal{O}_C$?

Motivation is clear: since $V_+(X)\cong \mathbb{P}^1$, we know it's cohomology and it we want to know about the genus of $C$ we need to understand the cohomology of the ideal sheaf $\mathcal{I}(V_+(\overline{X}))$ inside $C$.

Does such "Mayer-Vietoris approach" works in such situation the aim to compute the genus of $C$, or is for such kind of problems only the "normalization approach" helpful?

Let $C$ be be a connected reducible nodal curve over alg closed field $k$, such that all (finitely many) irred components $C_i$ of $C$ are smooth and intersections between different components are nodal singularities. (Mainly I'm interested in case $C$ is reducible stable curve.

Question: What are the standard techniques to calculate the genus $g_C:= H^1(C, \mathcal{O}_C)$?

Let denote $C= C_1 \cup C_2 \cup ... \cup C_n $. One technique which is familar to me is to normalize $f: \tilde{C} \to C$ and analyze the long cohomology sequence associated to exact seq

$$ 0 \to \mathcal{O}_C \to f_*\mathcal{O}_{\tilde{C}} \to \mathcal{F} \to 0 $$

Then by definition $\tilde{C}= \dot\bigcup_i ^n C_i $, so $g_{\tilde{C}}= \sum_{i=1}^n H^1(C_i, \mathcal{O}_{C_i}) $ (because after normalization the components become disjoint and all compt's were assumed to be smooth) and $\mathcal{F}$ has $0$-dimensional support, using this we get information about $g_C$. This should work.

But in this question I would like to find out if it's also possible to use inductive arguments (a la Mayer-Vietoris techniques) which appears more "natural" to me from geometrical point of view:

We can decompose $C$ as $D \cup C_n$ for $D:=C_1 \cup ... \cup C_{n-1}$. Let $\mathcal{I}(C_n), \mathcal{I}(D) \subset \mathcal{O}_C$ be the vanishing ideals wrt $C_n$ resp $D$ considered as subschemes on $C$. Then we obtain two natural sequences

$$ 0 \to \mathcal{I}(C_n) \to \mathcal{O}_C \to (i_{C_n})_*\mathcal{O}_{C_n} \to 0 $$

and

$$ 0 \to \mathcal{I}(D) \to \mathcal{O}_C \to (i_{D})_*\mathcal{O}_{D} \to 0 $$

Can we say something interesting about the cohomology of $\mathcal{I}(C_n)$ and $\mathcal{I}(D)$ cutting out whole components?

Let consider the most simple case I was able to think of: Say $C:= V_+(X \cdot Y) \subset \mathbb{P}^2= \operatorname{Proj}(k[X,Y,Z])$ is the "cross" with $C= V_+(X) \cup V_+(Y)$. In terms of above we have the sequence

$$ 0 \to \mathcal{I}(V_+(\overline{X})) \to \mathcal{O}_C \to (i_{V_+(X)})_*\mathcal{O}_{V_+(X)} \to 0 $$

where we consider $\overline{X}$ as homogeneous element in $ \in k[X,Y,Z]/(XY)$. Do we know something about cohomology of $\mathcal{I}(V_+(\overline{X})) $ regarded as ideal sheaf in $\mathcal{O}_C$?

Motivation is clear: since $V_+(X)\cong \mathbb{P}^1$, we know it's cohomology and it we want to know about the genus of $C$ we need to understand the cohomology of the ideal sheaf $\mathcal{I}(V_+(\overline{X}))$ inside $C$.

Does such "Mayer-Vietoris approach" work here to compute the genus of such type of curves, or is for such kind of problems only the "normalization approach" helpful?

Let $C$ be be a connected reducible nodal curve over alg closed field $k$, such that all (finitely many) irred components $C_i$ of $C$ are smooth and intersections between different components are nodal singularities. (Mainly I'm interested in case $C$ is reducible stable curve.

Question: What are the standard techniques to calculate the genus $g_C:= H^1(C, \mathcal{O}_C)$?

Let denote $C= C_1 \cup C_2 \cup ... \cup C_n $. One technique which is familar to me is to normalize $f: \tilde{C} \to C$ and analyze the long cohomology sequence associated to exact seq

$$ 0 \to \mathcal{O}_C \to f_*\mathcal{O}_{\tilde{C}} \to \mathcal{F} \to 0 $$

Then by definition $\tilde{C}= \dot\bigcup_i ^n C_i $, so $g_{\tilde{C}}= \sum_{i=1}^n H^1(C_i, \mathcal{O}_{C_i}) $ (because after normalization the components become disjoint and all compt's were assumed to be smooth) and $\mathcal{F}$ has $0$-dimensional support, using this we get information about $g_C$. This should work.

But in this question I would like to find out if it's also possible to use inductive arguments (a la Mayer-Vietoris techniques) which appears more "natural" to me from geometrical point of view:

We can decompose $C$ as $D \cup C_n$ for $D:=C_1 \cup ... \cup C_{n-1}$. Let $\mathcal{I}(C_n), \mathcal{I}(D) \subset \mathcal{O}_C$ be the vanishing ideals wrt $C_n$ resp $D$ considered as subschemes on $C$. Then we obtain two natural sequences

$$ 0 \to \mathcal{I}(C_n) \to \mathcal{O}_C \to (i_{C_n})_*\mathcal{O}_{C_n} \to 0 $$

and

$$ 0 \to \mathcal{I}(D) \to \mathcal{O}_C \to (i_{D})_*\mathcal{O}_{D} \to 0 $$

Can we say something interesting about the cohomology of $\mathcal{I}(C_n)$ and $\mathcal{I}(D)$?

Let consider the most simple case I was able to think of: Say $C:= V_+(X \cdot Y) \subset \mathbb{P}^2= \operatorname{Proj}(k[X,Y,Z])$ is the "cross" with $C= V_+(X) \cup V_+(Y)$. In terms of above we have the sequence

$$ 0 \to \mathcal{I}(V_+(\overline{X})) \to \mathcal{O}_C \to (i_{V_+(X)})_*\mathcal{O}_{V_+(X)} \to 0 $$

where we consider $\overline{X}$ as homogeneous element in $ \in k[X,Y,Z]/(XY)$. Do we know something about cohomology of $\mathcal{I}(V_+(\overline{X})) $ regarded as ideal sheaf in $\mathcal{O}_C$?

Motivation is clear: since $V_+(X)\cong \mathbb{P}^1$, we know it's cohomology and it we want to know about the genus of $C$ we need to understand the cohomology of the ideal sheaf $\mathcal{I}(V_+(\overline{X}))$ inside $C$.

Does such "Mayer-Vietoris approach" works in such situation the aim to compute the genus of $C$, or is for such kind of problems only the "normalization approach" helpful?

Let $C$ be be a connected reducible nodal curve over alg closed field $k$, such that all (finitely many) irred components $C_i$ of $C$ are smooth and intersections between different components are nodal singularities. (Mainly I'm interested in case $C$ is reducible stable curve.

Question: What are the standard techniques to calculate the genus $g_C:= H^1(C, \mathcal{O}_C)$?

Let denote $C= C_1 \cup C_2 \cup ... \cup C_n $. One technique which is familar to me is to normalize $f: \tilde{C} \to C$ and analyze the long cohomology sequence associated to exact seq

$$ 0 \to \mathcal{O}_C \to f_*\mathcal{O}_{\tilde{C}} \to \mathcal{F} \to 0 $$

Then by definition $\tilde{C}= \dot\bigcup_i ^n C_i $, so $g_{\tilde{C}}= \sum_{i=1}^n H^1(C_i, \mathcal{O}_{C_i}) $ (because after normalization the components become disjoint and all compt's were assumed to be smooth) and $\mathcal{F}$ has $0$-dimensional support, using this we get information about $g_C$. This should work.

But in this question I would like to find out if it's also possible to use inductive arguments (a la Mayer-Vietoris techniques) which appears more "natural" to me from geometrical point of view:

We can decompose $C$ as $D \cup C_n$ for $D:=C_1 \cup ... \cup C_{n-1}$. Let $\mathcal{I}(C_n), \mathcal{I}(D) \subset \mathcal{O}_C$ be the vanishing ideals wrt $C_n$ resp $D$ considered as subschemes on $C$. Then we obtain two natural sequences

$$ 0 \to \mathcal{I}(C_n) \to \mathcal{O}_C \to (i_{C_n})_*\mathcal{O}_{C_n} \to 0 $$

and

$$ 0 \to \mathcal{I}(D) \to \mathcal{O}_C \to (i_{D})_*\mathcal{O}_{D} \to 0 $$

Can we say something interesting about the cohomology of $\mathcal{I}(C_n)$ and $\mathcal{I}(D)$ cutting out whole components?

Let consider the most simple case I was able to think of: Say $C:= V_+(X \cdot Y) \subset \mathbb{P}^2= \operatorname{Proj}(k[X,Y,Z])$ is the "cross" with $C= V_+(X) \cup V_+(Y)$. In terms of above we have the sequence

$$ 0 \to \mathcal{I}(V_+(\overline{X})) \to \mathcal{O}_C \to (i_{V_+(X)})_*\mathcal{O}_{V_+(X)} \to 0 $$

where we consider $\overline{X}$ as homogeneous element in $ \in k[X,Y,Z]/(XY)$. Do we know something about cohomology of $\mathcal{I}(V_+(\overline{X})) $ regarded as ideal sheaf in $\mathcal{O}_C$?

Motivation is clear: since $V_+(X)\cong \mathbb{P}^1$, we know it's cohomology and it we want to know about the genus of $C$ we need to understand the cohomology of the ideal sheaf $\mathcal{I}(V_+(\overline{X}))$ inside $C$.

Does such "Mayer-Vietoris approach" works in such situation the aim to compute the genus of $C$, or is for such kind of problems only the "normalization approach" helpful?

Let $C$ be be a connected reducible nodal curve over alg closed field $k$, such that all (finitely many) irred components $C_i$ of $C$ are smooth and intersections between different components are nodal singularities. (Mainly I'm interested in case $C$ is reducible stable curve.

Question: What are the standard techniques to calculate the genus $g_C:= H^1(C, \mathcal{O}_C)$?

Let denote $C= C_1 \cup C_2 \cup ... \cup C_n $. One technique which is familar to me is to normalize $f: \tilde{C} \to C$ and analyze the long cohomology sequence associated to exact seq

$$ 0 \to \mathcal{O}_C \to f_*\mathcal{O}_{\tilde{C}} \to \mathcal{F} \to 0 $$

Then by definition $\tilde{C}= \dot\bigcup_i ^n C_i $, so $g_{\tilde{C}}= \sum_{i=1}^n H^1(C_i, \mathcal{O}_{C_i}) $ (because after normalization the components become disjoint) and $\mathcal{F}$ has $0$-dimensional support, using this we get information about $g_C$. This should work.

But in this question I would like to find out if it's also possible to use inductive arguments (a la Mayer-Vietoris techniques) which look more "natural" to me from geometrical point of view:

We can decompose $C$ as $D \cup C_n$ for $D:=C_1 \cup ... \cup C_{n+1}$. Let $\mathcal{I}(C_n), \mathcal{I}(D) \subset \mathcal{O}_C$ be the vanishing ideals wrt $C_n$ resp $D$ considered as subschemes on $C$. Then we obtain two natural sequences

$$ 0 \to \mathcal{I}(C_n) \to \mathcal{O}_C \to (i_{C_n})_*\mathcal{O}_{C_n} \to 0 $$

and

$$ 0 \to \mathcal{I}(D) \to \mathcal{O}_C \to (i_{D})_*\mathcal{O}_{D} \to 0 $$

Can we say something interesting about the cohomology of $\mathcal{I}(C_n)$ and $\mathcal{I}(D)$?

Let consider the most simple case I was able to think of: Say $C:= V_+(X \cdot Y) \subset \mathbb{P}^2= \operatorname{Proj}(k[X,Y,Z])$ is the "cross" with $C= V_+(X) \cup V_+(Y)$. In terms of above we have the sequence

$$ 0 \to \mathcal{I}(V_+(\overline{X})) \to \mathcal{O}_C \to (i_{V_+(X)})_*\mathcal{O}_{V_+(X)} \to 0 $$

where we consider $\overline{X}$ as homogeneous element in $ \in k[X,Y,Z]/(XY)$. Do we know something about cohomology of $\mathcal{I}(V_+(\overline{X})) $ regarded as ideal sheaf in $\mathcal{O}_C$?

Motivation is clear: since $V_+(X)\cong \mathbb{P}^1$, we know it's cohomology and it we want to know about the genus of $C$ we need to understand the cohomology of the ideal sheaf $\mathcal{I}(V_+(\overline{X}))$ inside $C$.

Does such "Mayer-Vietoris approach" works in such situation the aim to compute the genus of $C$, or is for such kind of problems only the "normalization approach" helpful?

Let $C$ be be a connected reducible nodal curve over alg closed field $k$, such that all (finitely many) irred components $C_i$ of $C$ are smooth and intersections between different components are nodal singularities. (Mainly I'm interested in case $C$ is reducible stable curve.

Question: What are the standard techniques to calculate the genus $g_C:= H^1(C, \mathcal{O}_C)$?

Let denote $C= C_1 \cup C_2 \cup ... \cup C_n $. One technique which is familar to me is to normalize $f: \tilde{C} \to C$ and analyze the long cohomology sequence associated to exact seq

$$ 0 \to \mathcal{O}_C \to f_*\mathcal{O}_{\tilde{C}} \to \mathcal{F} \to 0 $$

Then by definition $\tilde{C}= \dot\bigcup_i ^n C_i $, so $g_{\tilde{C}}= \sum_{i=1}^n H^1(C_i, \mathcal{O}_{C_i}) $ (because after normalization the components become disjoint and all compt's were assumed to be smooth) and $\mathcal{F}$ has $0$-dimensional support, using this we get information about $g_C$. This should work.

But in this question I would like to find out if it's also possible to use inductive arguments (a la Mayer-Vietoris techniques) which appears more "natural" to me from geometrical point of view:

We can decompose $C$ as $D \cup C_n$ for $D:=C_1 \cup ... \cup C_{n-1}$. Let $\mathcal{I}(C_n), \mathcal{I}(D) \subset \mathcal{O}_C$ be the vanishing ideals wrt $C_n$ resp $D$ considered as subschemes on $C$. Then we obtain two natural sequences

$$ 0 \to \mathcal{I}(C_n) \to \mathcal{O}_C \to (i_{C_n})_*\mathcal{O}_{C_n} \to 0 $$

and

$$ 0 \to \mathcal{I}(D) \to \mathcal{O}_C \to (i_{D})_*\mathcal{O}_{D} \to 0 $$

Can we say something interesting about the cohomology of $\mathcal{I}(C_n)$ and $\mathcal{I}(D)$?

Let consider the most simple case I was able to think of: Say $C:= V_+(X \cdot Y) \subset \mathbb{P}^2= \operatorname{Proj}(k[X,Y,Z])$ is the "cross" with $C= V_+(X) \cup V_+(Y)$. In terms of above we have the sequence

$$ 0 \to \mathcal{I}(V_+(\overline{X})) \to \mathcal{O}_C \to (i_{V_+(X)})_*\mathcal{O}_{V_+(X)} \to 0 $$

where we consider $\overline{X}$ as homogeneous element in $ \in k[X,Y,Z]/(XY)$. Do we know something about cohomology of $\mathcal{I}(V_+(\overline{X})) $ regarded as ideal sheaf in $\mathcal{O}_C$?

Motivation is clear: since $V_+(X)\cong \mathbb{P}^1$, we know it's cohomology and it we want to know about the genus of $C$ we need to understand the cohomology of the ideal sheaf $\mathcal{I}(V_+(\overline{X}))$ inside $C$.

Does such "Mayer-Vietoris approach" works in such situation the aim to compute the genus of $C$, or is for such kind of problems only the "normalization approach" helpful?