The correct spelling is ‘Kerala’. The word ‘Karela’ means a bitter gourd (vegetable) in a certain language in India.

@WillSawin yes, admittedly, it did sound silly, I’ll just leave it there for now. I’m not sure how accurate this reference is, a link with an example of finite rings math.stackexchange.com/questions/530591/…

Of course, we could take the product of F_p and F_q, which just translates to disjoint union of the Spec but that’s a silly example and I’m not sure if you want that.

Have you seen Batyrev's result? arxiv.org/abs/alg-geom/9310003

I think the OP is thinking of some specific K3 surface or a degeneration of (2,3)-K3 surfaces.

Is the assertion that $ (I_Y)^{\vee} = \mathcal{O}_X $ a consequence of the following steps: (1) $ (I_Y)^{\vee} $ is reflexive by a short lemma (2) $ (I_Y)^{\vee \vee} = \mathcal{O}_X $ because the double dual is a line bundle which is trivial outside a codimension $ 2 $ closed subset, hence is trivial by Hartog's lemma. I don't immediately see how you get the first dual trivial and then the second.

In a special case where $ X $ is a curve, $ \operatorname{Hom}(E, E \otimes \omega_X) \cong \operatorname{Ext}^1(E,E)^{\vee} $ by Serre duality. Therefore Higgs sheaves on $ X $ correspond to the cotangent bundle of the moduli stack of vector bundles on $ X $.

(1) The total space needs to have a dual on the line bundle $ \mathcal{L} $. (2) The normal bundle of the zero section of C in L is $ \mathcal{L} $ itself - this is intuitively obvious from a mental picture.

I’m not sure whether this guarantees that the scheme-theoretic intersection is X. (Set-theoretically, it does)

I have a question on the side: Can the Picard number of a quartic K3 be any possible integer between 1 and 20 inclusive?

@JasonStarr Oh wow. This maximal open set U will have complement codimension 2 or larger - because the local ring at the generic point of a codimension 1 subvariety is of dimension 1 hence a DVR so torsion-free stalk of F is the same as free stalk of F there. Now on that maximal open set, the definitions of det agree and the restriction Pic X to Pic U is an isomorphism because of codimension >=2 again. Correct?

I have a related question. Is it true that for a general point p = [C] in M_g, we have that the endomorphism ring of Jac(C) is Z?

@JasonStarr I understand. When I first phrased the question, it was confusing but now it’s absolutely clear once I wrote it down.