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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?

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    $\begingroup$ In your notation $\mathcal{I}(C_n)=\mathcal{O}_D(-\Delta)$ where $\Delta=C_n\cap D$, then you should be able to say something about those cohomologies. Working in this way you can prove a famous formula by Hironaka which computes the arithmetic genus of a reducible curve, see the answear by Georges Elencwajg in this post math.stackexchange.com/questions/1297433/…. Using this one you can argue by induction on the number of components, supposing as you do $C=C_n\cup D$. $\endgroup$
    – Wiktor Vacca
    Commented Dec 16, 2023 at 11:54
  • $\begingroup$ two points in your explanations are not clear to me. Firstly, why we can identify $\mathcal{I}(C_n)=\mathcal{O}_D(-\Delta)$, assuming we using notations from above where the ideal sheaf $\mathcal{I}(C_n)$ is defined to sit in ses $0 \to \mathcal{I}(C_n) \to \mathcal{O}_C \to (i_{C_n})_*\mathcal{O}_{C_n} \to 0$? $\endgroup$
    – user267839
    Commented Dec 17, 2023 at 15:38
  • $\begingroup$ secondly, do I understand your concern correcty, that when you wrote "...then you should be able to say something about those cohomologies." comes from inductive reasoning; namely we can say something about cohom of $\mathcal{O}_D(-\Delta)$ based on assuming we inductively "understood" cohom of $\mathcal{O}_D$ and now analyze the assoc long ex seq containing cohom of $\mathcal{O}_D(-\Delta)$? Or did I misunderstood your suggested strategy? $\endgroup$
    – user267839
    Commented Dec 17, 2023 at 15:45
  • $\begingroup$ To put it in a nitshell, I'm wondering if when you say that one can say something about cohom of $\mathcal{O}_D(-\Delta)$ at that stage , if you refering there exactly to the amount of information about it extractible from "natural" les associated to ses $0 \to \mathcal{O}_D(-\Delta) \to \mathcal{O}_D \to \mathcal{O}_{\Delta} \to 0$ only, or do you refer in your comment above to another argument in order to gain information one can gain about cohom of $\mathcal{O}_D(-\Delta)$? $\endgroup$
    – user267839
    Commented Dec 17, 2023 at 15:57
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    $\begingroup$ The sections of the kernel of $\mathcal{O}_C \to (i_{C_n})_*\mathcal{O}_{C_n} \to 0$ are functions on $C$ vanishing on $C_n\cap D=\Delta$, this gives you a function on $D$ vanishing on $C_n$, hence a section of $\mathcal{O}_D(-\Delta)$, you can check that this defines actually an isomorphism of sheaves. For the second question: it depends on what you know about the curves. What I had in mind was the sequence you wrote. $\endgroup$
    – Wiktor Vacca
    Commented Dec 21, 2023 at 15:19

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