Tangent spaces of orbit closures for representations of Dynkin quivers of type D

Let $\Bbbk$ be an algebraically closed field, $Q$ a finite quiver, and denote by $\mathop{\mathrm{rep}}_Q^{\mathbf{d}}$ the affine $\Bbbk$-scheme of representations of $Q$ with a fixed dimension vector ${\mathbf{d}}$. Given a representation $M$ of $Q$ with dimension vector ${\mathbf{d}}$, the set ${\mathcal{O}}_M$ of points in $\mathop{\mathrm{rep}}_Q^{\mathbf{d}}(\Bbbk)$ isomorphic as representations to $M$ is an orbit under an action on $\mathop{\mathrm{rep}}_Q^{\mathbf{d}}(\Bbbk)$ of a product of general linear groups. The orbit ${\mathcal{O}}_M$ and its Zariski closure $\overline{{\mathcal{O}}}_M$, considered as reduced subschemes of $\mathop{\mathrm{rep}}_Q^{\mathbf{d}}$, are contained in an affine scheme ${\mathcal{C}}_M$ defined by rank conditions on suitable matrices associated to $\mathop{\mathrm{rep}}_Q^{\mathbf{d}}$. For all Dynkin and extended Dynkin quivers, the sets of points of $\overline{{\mathcal{O}}}_M$ and ${\mathcal{C}}_M$ coincide, or equivalently, $\overline{{\mathcal{O}}}_M$ is the reduced scheme associated to ${\mathcal{C}}_M$. Moreover, $\overline{{\mathcal{O}}}_M={\mathcal{C}}_M$ provided $Q$ is a Dynkin quiver of type ${\mathbb{A}}$, and this equality is a conjecture for the remaining Dynkin quivers (of type ${\mathbb{D}}$ and ${\mathbb{E}}$). Let $Q$ be a Dynkin quiver of type ${\mathbb{D}}$ and $M$ a finite dimensional representation of $Q$. We show that the equality $T_N\overline{{\mathcal{O}}}_M=T_N{\mathcal{C}}_M$ of Zariski tangent spaces holds for any closed point $N$ of $\overline{{\mathcal{O}}}_M$. As a consequence, we describe the tangent spaces to $\overline{{\mathcal{O}}}_M$ in representation theoretic terms.


Introduction and the main results
Throughout the paper k denotes an algebraically closed field of arbitrary characteristic.We will identify a k-scheme X with its functor of points, i.e. the functor from the category of commutative k-algebras to the category of sets sending R to the set of morphisms Spec(R) → X .Let k[ε] denote the k-algebra of dual numbers and consider the map where π : k[ε] → k is the canonical surjective homomorphism.Given a k-rational point x of X , i.e. x ∈ X (k), the fiber X (π) −1 (x) is the Zariski tangent space T x X to X at x.We are mostly interested in affine schemes X of finite type over k, i.e. the schemes of the form Spec(R), where R is a finitely generated commutative k-algebra.For such schemes X (k) is the set of closed points of X .
Let Q = (Q 0 , Q 1 ) be a finite quiver, i.e. a finite set Q 0 of vertices and a finite set Q 1 of arrows α : sα → tα, where sα and tα denote the starting and the terminating vertex of α, respectively.A representation of Q over k is a collection M = (M a , M α ; a ∈ Q 0 , α ∈ Q 1 ) of k-vector spaces M a and k-linear maps M α : M sα → M tα .A morphism f : M → N between two representations is a collection f = (f a : M a → N a ; a ∈ Q 0 ) of k-linear maps such that We denote by rep(Q) the category of finite dimensional representations of Q, i.e. the representations M such that all vector spaces M a are finite dimensional.For a representation M in rep(Q) we define its dimension vector dim M = (dim k M a ) ∈ N Q 0 .We denote by M p,q the k-scheme of p × q-matrices and by GL d the group k-scheme of invertible d × d-matrices, for any positive integers p, q and d.Given a dimension vector d = (d a ) ∈ N Q 0 we have the affine scheme Thus the points of rep d Q (k) can be identified with the representations M of Q such that M a = k da for any a ∈ Q 0 .The group scheme (see for instance [11]).Moreover, due to Bongartz [3,4], the reverse implication holds under an additional assumption on Q.
Theorem 1.1.Let Q be a Dynkin or an extended Dynkin quiver.Assume M and N belong to rep(Q) and dim M = dim N .Then M degenerates to N if and only if the condition (1.1) is satisfied.
Inspired by the above inequalities (see also [3,Proposition 1]), a closed GL d -subscheme C M of rep d Q containing O M was defined in [12].Let kQ = a,b∈Q 0 kQ(a, b) denote the path algebra of Q, where kQ(a, b) is the vector space with a k-basis formed by the paths in Q starting at b and terminating at a.For any commutative k-algebra R, X ∈ rep d Q (R) and ω ∈ kQ(a, b), the matrix X ω ∈ M da,d b (R) is defined in the obvious way.
Let p, q ∈ N, and consider two sequences (a 1 , . . ., a p ) and (b 1 , . . ., b q ) of vertices in Q 0 and a p × q-matrix ω = (ω i,j ) such that each ω i,j belongs to kQ(a i , b j ).We define a regular morphism , where ω runs through all possible matrices of linear combinations of paths with all possible sequences of starting and terminating vertices.Then ). Hence we can reformulate Theorem 1.1 by saying that C M (k) = O M (k) provided Q is a Dynkin or an extended Dynkin quivers.Note that this equality does not hold for the representation Here C M (k) has two irreducible components of dimension 5, with one of them being O M (k) (see [12,Example 8.9]).
Lakshmibai and Magyar described in [8] generators of the defining ideal of O M in case Q is an equioriented Dynkin quiver of type A. They used the Zelevinsky immersion of rep d Q in a Schubert variety of a flag variety, and applied a description of generators of the defining ideal of this Schubert variety.It turned out that they showed that the defining ideal of O M equals I M .The result was generalized in [12] to the Dynkin quivers of type A with an arbitrary orientation: We do not know if this result remains true for the Dynkin quivers of type D (D n , n ≥ 4) and E (E 6 , E 7 , E 8 ).It would be interesting to know if C M and O M have at least identical tangent spaces in these cases.Note that this is not true even for the simplest extended Dynkin quiver.Namely, consider representations We give now a representation theoretic interpretation of ) have natural structures of vector spaces over k.Using the decomposition k . We remark that Theorem 1.3 should have applications in the problem of describing the singular locus of O M in representation theoretic terms, for the representations M of the Dynkin quivers of type D (see [12,Section 8]).
The paper is organized as follows: in Section 2 we prove necessary facts about exact sequences in rep(Q) (in fact we formulate them in a more general setup of triangles in the derived category of rep(Q)), while in Section 3 we apply results of Section 2 in geometric context and prove the main result.For basic background on representation theory of quivers we refer to [1,2,13].
The both authors gratefully acknowledge the support of the National Science Centre grant no.2020/37/B/ST1/00127.

Derived categories for representations of Dynkin quivers
In order to prove Theorem 1.3, we need a result about existence of short exact sequences in rep(Q) with some special properties, where Q is a Dynkin quiver of type D (Corollary 2.23).Our idea is to use the embedding of rep(Q) in its derived category D b (Q) = D b (rep(Q)), and to prove existence of triangles in D b (Q) satisfying similar properties (Proposition 2. 19).An advantage of working with the derived category is that its structure, including Auslander-Reiten theory, is more "regular" than the structure of rep(Q).We refer to [7] as a general reference for this section.

Dynkin graphs
Throughout this section ∆ = (∆ 0 , ∆ 1 ) is a Dynkin graph of one of the types A, D or E: where ∆ 0 is the set of n vertices of ∆, and ∆ 1 is its set of edges, i.e. two element subsets of ∆ 0 .If {a, b} is an edge we say that a and b are adjacent.We denote by a − the (open) neighbourhood of a vertex a, i.e. the set of vertices adjacent to a.The degree of a equals, by definition, the cardinality of a − .Let dist(a, b) be the length of the shortest walk in ∆ between a and b.
We define an integer n ∆ as follows: With ∆ we associate (Tits) quadratic form which is positive definite, i.e. q ∆ (d) > 0 for any non-zero d.If q ∆ (d) = 1 we say d is a root.Obviously, if d is a root, then −d is also a root.There are n •n ∆ roots, half of them are positive, where a vector d is called positive provided d = 0 and d a ≥ 0, for each a.There is a unique maximal root h ∆ (i.e.h ∆ − d is positive for any root d = h ∆ ) which equals

Derived category for acyclic quivers
Throughout this subsection Q is a finite quiver without oriented cycles.We denote by D b (Q) = D b (rep(Q)) the derived category of the abelian category rep(Q).The category D b (Q) is triangulated, hence there is an auto-equivalence [1] of D b (Q) called the shift functor ("the suspension functor" and "the translation functor" are alternative names used by other authors) and a class of triangles (the name "distinguished triangles" is commonly used), written in the form . There is a canonical full embedding of rep(Q) in D b (Q), and we shall identify rep(Q) with its image in D b (Q).In particular, for all X, Y ∈ rep(Q).Based on the latter equality, there is a strong relationship between the short exact sequences in rep(Q) and triangles in D b (Q).Namely, for each short exact sequence . We generalize now to triangles notion of a split exact sequence and a pullback.Let σ be a triangle Observe that an exact sequence σ splits if and only if the triangle σ splits.

Given a triangle
where the upper row, called the pullback of σ along h, is a triangle (note that the pullback is unique up to isomorphism of triangles).One defines pushouts dually.Observe that if σ ′ is the pullback of a short exact sequence σ along a homomorphism h : We say that a triangle σ is almost split (or an Auslander-Reiten triangle) if A and C are indecomposable, σ does not split, but its pullbacks split for all morphisms to C which are not retractions.The last condition can be replaced by the requirement that the pushouts of σ split for all morphisms from A which are not sections.
An important fact about the category D b (Q) is that it has a Serre duality, i.e. there is an auto-equivalence ν : D b (Q) → D b (Q), called a Serre functor, such that there are isomorphisms which are natural in X and Y , where D is the duality Hom k (?, k) on mod k (see [9, I]).The Serre functor ν restricts to an equivalence between the subcategory P Q of the projective representations in rep(Q) and the subcategory I Q of the injective representations in rep(Q).This restriction is called a Nakayama functor.For each vertex a of Q, we denote by P a and I a the indecomposable projective and injective representation in rep(Q) at a, respectively.We note that up to isomorphism, these are the only indecomposable objects of P Q and I Q , respectively.The existence of a Serre functor ν is closely related to the existence of almost split triangles in D b (Q).Namely, we consider the auto-equivalence τ and call it the Auslander-Reiten translation.Then there is an almost split triangle of the form τ C → B → C → (τ C) [1] for any indecomposable object C in D b (Q), and there is an almost split triangle of the form One defines the Grothendieck group K 0 (rep(Q)) of rep(Q) as the quotients of the free abelian group with basis formed by the isomorphism classes [X] of objects X in rep(Q), modulo the subgroup generated by the map sending the class of [X] to dim X.We will treat this isomorphism as an identification.
The Grothendieck group K 0 (D b (Q)) of the category D b (Q) is defined in a similar way, the only difference is that one replaces the sequences 0 , which we will treat as identification.In particular we will use notation dim Observe that the quadratic form associated with b Q coincides with q ∆ , where ∆ is the underlying graph of Q and q ∆ is the quadratic form introduced in (2.2).Recall that Yoneda lemma states We collect below few facts concerning indecomposable objects in D b (Q) under the assumption that Q is a Dynkin quiver.Lemma 2.1.Let Q be a Dynkin quiver, and X and Y be indecomposable objects in D b (Q).Then: Proof.The first three properties are well known (see for example [7]).( 4).We may assume that [X, Y ] > 0. Applying the automorphism τ −n X we get where the last equality follows from (3) and the assumption [X, Y ] > 0. As we observed above is a root, and the claim follows, since h ∆ is the maximal root.

Mesh categories for Dynkin graphs
Throughout this subsection ∆ = (∆ 0 , ∆ 1 ) is a Dynkin graph.We say that two elements (p, a) and (q, b) of the product Z × ∆ 0 are equivalent provided the integer (q − p) + dist(a, b) is even.This is an equivalence relation since ∆ is a tree, and thus Z × ∆ 0 is partitioned into two parts In order to decide which part stands for (Z × ∆ 0 ) v , we choose a base vertex b 0 ∈ ∆ 0 and require that (0, b 0 ) belongs to (Z × ∆ 0 ) v .We define an infinite quiver Z∆ without multiple arrows as follows.The set (Z∆) 0 of vertices of Z∆ consists of v p,a , where (p, a) ∈ (Z × ∆ 0 ) v .There is an arrow in Z∆ starting at v p,a and terminating at v q,b if and only if a and b are adjacent in ∆ and q − p = 1.For example, if and we choose b 0 = b, then Z∆ has the form For each pair (p, a) in (Z × ∆ 0 ) m , we consider the smallest subquiver m p,a of Z∆, called a mesh, containing all paths (of length two) starting at v p−1,a and terminating at v p+1,a : where {b 1 , . . ., b r } = a − .
We denote by (Z∆) 2 the set of meshes m p,a , where (p, a) ∈ (Z × ∆ 0 ) m .With each mesh m p,a we associate its mesh relation, i.e. the sum of the paths starting at v p−1,a and terminating at v p+1,a , considered as morphisms in the path category k[Z∆] of Z∆.The mesh category k(Z∆) of Z∆ is the quotient of k[Z∆] modulo the ideal generated by all mesh relations.Now let Q be a Dynkin quiver with underlying graph ∆.An important fact is that k(Z∆) is equivalent as a k-linear category to the category of indecomposable objects in D b (Q).When Q is fixed, then we shall identify (Z∆) 0 with a complete set of pairwise non-isomorphic indecomposable objects of D b (Q).Moreover, we may also assume that under this identification P a = v pa,a with an appropriate integer p a , for each vertex a ∈ Q 0 = ∆ 0 .The three crucial auto-equivalences of D b (Q): the Auslander-Reiten translation τ , the Serre functor ν and the shift functor [1] act on the indecomposable objects by the formulas where n ∆ was defined in (2.1) and φ ∆ is the automorphism of ∆ defined as follows: φ ∆ is the unique non-trivial involution of ∆ provided ∆ is either of type A n with n ≥ 2, or D n with n odd, or E 6 ; and φ ∆ is the identity on ∆ for the remaining Dynkin graphs.We note that the automorphism of Z∆ induced by ν is sometimes called a Nakayama permutation ([6, 6.5]).We also remark that the quiver Z∆ is isomorphic to the quiver ZQ defined in [10].
The almost split triangles in D b (Q) are parameterized by the meshes in Z∆.More precisely, there is an almost split triangle of the form AR(m p,a ) : We note that the paths in Z∆ have the form where each two consecutive vertices in the sequence (a p , a p+1 , . . ., a q−1 , a q ) are adjacent.Therefore ω can be viewed as a lifting of a walk in ∆.In particular, q − p ≥ dist(a p , a q ).The path ω is called sectional if a i−1 = a i+1 for any integer i with p < i < q.Since ∆ is a tree, this condition is equivalent to the fact that the vertices a p , . . ., a q are pairwise different, and also equivalent to the equality q − p = dist(a p , a q ).Lemma 2.2.Let v p,a and v q,b be vertices in Z∆.Then: (1) [v p,a , v p,a ] = 1. ( (1) follows from Lemma 2.1 (2), but can also be derived directly from the definition of the mesh category k(Z∆).( 2) is a consequence of the Serre duality (2.3). ( . Hence there are paths in Z∆ from v p,a to v q,b and from v q,b to v p+n ∆ −2,φ ∆ (a) .
(4).Since P a = v pa,a , v p,a = τ r P a for some integer r.
b follows from the first one and (2).Let us explain how using the above lemma and almost split sequences, we can calculate the dimension [v p,a , v q,b ] for all vertices v p,a and v q,b , Namely, if q ≤ p or q ≥ p + n ∆ − 1 then [v p,a , v q,b ] = 0 except [v p,a , v p,a ] = 1.We obtain formulas in the remaining cases by induction on q, using the following lemma.
Lemma 2.3.Let v p,a and v q,b be vertices in Z∆ such that p < q < p + n ∆ .Then Proof.Applying the functor Hom D b (Q) (v p,a , ?) to the triangle AR(m q−1,b ) we get the exact sequence The two extreme homomorphism spaces are zero by Lemma 2.2(3), and the claim follows.
Applying the above to sectional paths we get the following.
Proof.The claim follows by induction on the length (q − p) of the path, where the base step q − p = 0 follows from Lemma 2.2.For the induction step we apply Lemma 2.3 for v p,a = v p,ap and v q,b = v q,aq , and use that there is no path in Z∆ from v p,ap to v q−2,aq , and if there is a path from v p,ap to v q−1,c with c ∈ (a q ) − then c = a q−1 .

Defect functions on meshes
Throughout this subsection Q is a Dynkin quiver with the underlying graph ∆.In paricular, and an object X in D b (Q).There is a commutative diagram of the form where the vertical arrows represent k-linear isomorphisms obtained by applying the Auslander-Reiten translation τ and the Serre duality (2.3).This inspires to define the following integervalued function measuring how far is a triangle from being split.Definition 2.5.Given a triangle σ : Some fundamental properties of δ σ being easy consequences of the definition are collected in the following lemma.
Lemma 2.6.Let σ : Then the following hold: (1) σ splits if and only if δ σ = 0. (2) Example 2.7.Let ∆ be the Dynkin graph D 6 : Thus n ∆ = 10 and the quiver Z∆ has the form We consider the triangle σ : , by Lemma 2.6 (2).We find δ σ by calculating the dimensions [v 3,b 1 , v q−1,b ], which can be done by the method based on Lemma 2.3.We illustrate the function δ σ by writing each non-zero value δ σ (m p,a ) between the vertices v p−1,a and v p+1,a : We set X, Y = i≤0 (−1) i • [X, Y ] i for any objects X and Y in D b (Q).Our next aim is to define integer-valued functions on the set of meshes, using X, Y .We derive from Lemma 2.2(3) the following fact.
By Corollary 2.8 we conclude the following fact.Applying Hom functors we get the following.
Corollary 2.12.δ σ = δ B,A⊕C for any triangle σ : Combining the above corollary and Lemma 2.6(5) we get the following fact.Lemma 2.13.Let (p, a) and (q, b) belong to Let N and X be objects of D b (Q) and assume that X is indecomposable.We denote by mult X (N ) the multiplicity of X as a direct summand of N .In particular, As an immediate consequence of Lemma 2.13 we get: Applying the above corollary for the vertices lying on a sectional path we obtain the following fact.
be a sectional path in Z∆.Let M be the subset of (Z × ∆ 0 ) m consisting of pairs (j, b) such that p ≤ j ≤ q and b is adjacent to a j , but does not belong to the set {a p , . . ., a q }.Then Observe in the above situation that if b is adjacent to a j , then b belongs to the set {a p , . . ., a q } if and only if either j > p and b = a j−1 or j < q and b = a j+1 .

Application to type D
Throughout this subsection Q is a Dynkin quiver of type D n , n ≥ 4, with the underlying graph ∆: In particular, h ∆ a = 1 if a ∈ {c, c ′ , c ′′ }, and h ∆ a = 2 otherwise.Applying Lemmas 2.1(4) and 2.6(3), we get the following corollaries.
Corollary 2.17.[v p,a , v q,b ] ≤ 2 and the inequality is strict if at least one of the vertices a and b belongs to {c, c ′ , c ′′ }.
The main aim of this subsection is to prove the following fact.
Then there is an indecomposable direct summand C ′ of N together with a morphism h : We introduce two integer-valued functions ϕ and ψ on (Z∆) 0 ⊔ (Z∆) 2 as the compositions of the canonical bijection (Z∆) 0 ⊔ (Z∆) 2 → Z × ∆ 0 followed by the maps We can illustrate the above statement about the function δ σ for r = 2 (hence n ≥ 6) by the following picture Proof.The claim follows by induction on r ≥ 0 from the following two properties of δ σ : Indeed, the base step follows from (ii) and the induction step from (i).
Combining the assumption δ σ (m p 0 ,br ) = 2 with Lemma 2.6(3) and Corollary 2.17 gives the equalities Let L be the set of vertices lying on the following sectional path in Z∆: On the other hand, by Corollary 2.18, δ σ (m Consequently, we get three equalities of the six equalities appearing in (i) and (ii).
Dual considerations for the following sectional path in Z∆: lead to the remaining three equalities in (i) and (ii).
Proof of Proposition 2.19.Let m p 0 ,a 0 be a mesh satisfying δ M,N (m p 0 ,a 0 ) < δ σ (m p 0 ,a 0 ).By Corollary 2.11, we can choose such a mesh having the minimal value ψ(m p 0 ,a 0 ).We conclude from the assumption supp(δ σ ) ⊆ supp(δ M,N ) and Corollary 2.18 that δ M,N (m p 0 ,a 0 ) = 1, δ σ (m p 0 ,a 0 ) = 2 and a 0 = b r for some r with 0 ≤ r ≤ n − 4. Let R be the subset of (Z∆) 0 consisting of vertices v p,a such that ϕ(v p,a ) > ϕ(m p 0 ,br ) and ψ(v p,a ) < ψ(m p 0 ,br ).We illustrate the set R for r = 3 (hence n ≥ 7) using 14 big dots: The key observation is that by Corollary 2.15 we get the formula vp,a∈R Combining Lemma 2.20 with the assumption supp(δ σ ) ⊆ supp(δ M,N ) and using δ M,N (m p 0 ,br ) = 1, we get that the right-hand side is at least 2. Hence mult C ′ (N ) > 0 for some Again by Lemma 2.20, δ σ (m p ′ −1,a ′ ) > 0, which from the definition of δ σ means that γ • h = 0 for some morphism h : C ′ → C. Let σ ′ be the pullback of σ along h.We need to prove that Since δ σ ′ ≤ δ σ and supp(δ σ ) ⊆ supp(δ M,N ), it suffices to show that δ M,N (m p,a ) ≥ δ σ (m p,a ) whenever δ σ ′ (m p,a ) > 0. Thus we assume that δ σ ′ (m p,a ) > 0. By Lemma 2.6(3), [v p+1,a , C ′ ] > 0, and from Lemma 2.2(3) we conclude that ψ(v p+1,a ) ≤ ψ(C ′ ).Using the fact that C ′ belongs to R and how the latter was defined, we get the following sequence of inequalities It follows from our choice of the mesh m p 0 ,br that δ M,N (m p,a ) ≥ δ σ (m p,a ), which finishes the proof.

Passage from
The main aim of this subsection is to prove a result analogous to Proposition 2.19, concerning the category rep(Q), where Q is a Dynkin quiver of type D. Throughout this subsection Q is a Dynkin quiver with its underlying graph ∆.
As observed in Subsection 2.3, the k-linear structure of the category D b (Q) is fully described by the quiver Z∆.Similarly, the category rep(Q) is fully described by its Auslander-Reiten quiver Γ Q .Moreover, the identification of rep(Q) as a full subcategory of D b (Q) correspond to the identification of Γ Q as a full convex subquiver of Z∆, which we are going to explain.We note that introducing the Auslander-Reiten quiver Γ Q as a subquiver of Z∆ was done already in [6, 6.5].
By a slice in Z∆ we mean a full convex subquiver containing exactly one vertex v ra,a for each a ∈ ∆ 0 .Thus |r a − r b | = 1 for any adjacent vertices a and b.Recall that P a = v pa,a for any vertex a ∈ Q 0 = ∆ 0 .The vertices P a , a ∈ Q 0 , together with the arrows connecting them form a slice S isomorphic to Q op , where Q op is the opposite quiver of Q having the same set of vertices, but with the arrows reversed.Consequently, the vertices I a = ν(P a ) = v pa+n ∆ −2,φ ∆ (a) , a ∈ Q 0 , lie on a slice νS, which is also isomorphic to Q op .Then the Auslander-Reiten quiver Γ Q is the smallest full convex subquiver of Z∆ containing S and νS.We denote by (Γ Q ) 2 the set of all meshes in Z∆ which are contained in Γ Q .
The shifts (Γ Q )[i], i ∈ Z, are pairwise disjoint subquivers of Z∆, hence we have the following inclusion In fact, this inclusion is the equality on the sets of vertices, and only the arrows connecting νS[i] with S[i + 1], i ∈ Z, are missing (see for instance [7, 5.5]).For example, if Q is the quiver then the above embedding of quivers looks as follows.
Let X and Y be representations in rep(Q).Then [X, Y ] i = 0 for i < 0, and hence X, Y = [X, Y ].Therefore the definition of the integer-valued function δ M,N simplifies if we restrict to the subcategory rep(Q) as the following result explains.Lemma 2.21.Let Q be a Dynkin quiver.Assume that M and N belong to rep(Q) and dim M = dim N .Let m p,a be a mesh in (Z∆) 2 .Then if m p,a belongs to (Γ Q ) 2 , and δ M,N (m p,a ) = 0, otherwise.
Recall that for exact sequence σ : 0 We set δ σ = δ σ .In this case Definition 2.5 and Corollary 2.12 simplify as shown in the following lemma.We also note that the function δ σ is closely related to the defect functors considered in [1,IV.4].
) if and only if the short exact sequence σ(U, Z, V ) splits.The quotient of Z 1 Q (V, U ) by B 1 Q (V, U ) can be identified with the extension group Ext 1 Q (V, U ) of V by U , and we have the following exact sequence Applying the above for U = N = V , where N ∈ rep d Q (k), we get the exact sequence The space V k d k d (k) can be identified with the tangent space T 1 GL d , the space Q , and η N,N with the tangent map induced by the orbit map Under this identification, , which is a famous Voigt result [14].

Proof of the main result
Throughout this subsection Q is a Dynkin quiver, M a representation in rep(Q) and d = dim M .We will work with closed subschemes of rep d Q and hence we start with a few general remarks on subschemes.If X is a subscheme of a k-scheme Y, then the corresponding map X (R) → Y(R) is injective, and we will identify X (R) with its image in Y(R), for any commutative k-algebra R. The following fact can be concluded from [5, I.2.6.1]:Lemma 3.1.Let X be a closed subscheme of a k-scheme Y and ϕ : R → S an injective homomorphism of commutative k-algebras.Then X (R) = Y(ϕ) −1 (X (S)).
We will use several times the above lemma for a k-scheme Y which is affine (specifically for Y = rep d Q ).Then the closed embedding X ⊆ Y is isomorphic to Spec(ψ) : Spec(A/I) → Spec(A), where ψ : A → A/I is the canonical surjective homomorphism, for some commutative k-algebra A and its ideal I. Hence the claim translates to an obvious fact about the existence of a homomorphism completing a given commutative diagram in the category of k-algebras to another commutative diagram, as follows: If ϕ : R → S is an injective k-algebra homomorphism, X is a k-scheme and x ∈ X (R), it will be convenient to denote the image of x under the map X (ϕ) : X (R) → X (S) also by x.
Let π : k[ε] → k denote the canonical surjective homomorphism.Since O M is a closed subscheme of C M and the latter is a closed subscheme of rep d Q , we have the following commutative diagram with inclusions: where the equality in the second row follows from (a reformulation of) Theorem 1 In particular, . Assume that we have a fixed decomposition N = s∈S N s as a representation of Q.In particular, the collection (N a ) = k d decomposes as s∈S (N s a ).If p, q ∈ S, then using notation introduced after (3.2) we have H p,q ∈ A N q N p (k) for each H ∈ A k d k d (k).Equality (3.4) implies that we can apply the above notation both when H = L is a point of rep d Q (k), and when H = Z is viewed as tangent vector.Conversely, given H p,q ∈ A N q N p (k) we have H p,q ∈ A k d k d (k).Observe that N p,p is the collection (N p α ) and N p,q = 0, for all p = q in S, hence N = s∈S N s,s .Analogously, applying notation introduced after (3.2) for the scheme V k d k d (k) we also have g p,q ∈ V N q N p (R) for g ∈ GL d (R).Finally, if s ∈ S, we set 1 s for the identity on N s , which is an element of ) and two indices p = q in S. Then the following conditions hold: (1) δ L,N = δ σ(N p ,Z p,q ,N q ) .
(3) L ≃ N if and only if Z p,q belongs to B 1 Q (N q , N p ).
(3).We have from (1) that L is isomorphic to N if and only if the sequence σ(N p , Z p,q , N q ) splits.The latter means that Z p,q belongs to B 1 Q (N q , N p ).
Lemma 3.3.Let N = s∈S N s be a fixed decomposition of a point N in O M (k).Let Z p,q ∈ A N q N p (k), for p = q in S, be such that δ σ(N p ,Z p,q ,N q ) ≤ δ M,N .Then N + Z p,q ∈ O M (k) and N + ε • Z p,q ∈ O M (k[ε]) (equivalently, Z p,q ∈ T N O M ).
We claim that the point N + t • Z p,q of rep d Q (k[t]) belongs to O M (k[t]).Consider the element g in GL d (k[t, t −1 ]) given by g = s =p 1 s + t • 1 p .Obviously g −1 = s =p 1 s + t −1 • 1 p .Since L = s∈S N s,s + Z p,q , g ⋆ L = g • L • g −1 = s∈S N s,s + t • Z p,q = N + t • Z p,q as elements of rep Applying the homomorphism k[t] → k[ε] sending t to ε we get that N + ε • Z p,q belongs to O M (k[ε]).
The above lemma gives a method of detecting vectors tangent to O M .This method is sufficient for the representations of the Dynkin quivers of type A as the proposition below shows.Obviously this proposition follows also immediately from Theorem 1.2.Proof.Let N ∈ O M (k) and fix a decomposition N = N s of N such that each N s is indecomposable.
Choose Z ∈ T N C M = Z 1 M,N (N, N ).Since Z = p,q∈S Z p,q (see (3.3)), it is sufficient to show that Z p,q ∈ T N O M , for all p, q ∈ S. Fix such p and q.We may assume Z p,q ∈ B 1 Q (N, N ), as Since Z 1 M,N (?, ?) and B 1 Q (?, ?) are subfunctors of Z 1 Q (?, ?), Z p,q belongs Z 1 M,N (N q , N p ) but not to B 1 Q (N q , N p ).In particular, Ext 1 Q (N q , N p ) is non-zero.Since Q is a Dynkin quiver, N q is not isomorphic to N p , hence q = p.
It follows from the definition of Z 1 M,N (?, ?) that supp(δ σ(N p ,Z p,q ,N q ) ) ⊆ supp(δ M,N ).Combining Lemmas 2.2(4) and 2.6(3) we get that the values of the function δ σ(N p ,Z p,q ,N q ) do not exceed 1. Hence we conclude the inequality δ σ(N p ,Z p,q ,N q ) ≤ δ M,N , thus Z p,q belongs to T N O M , by Lemma 3.3.
The above method does not extend to the Dynkin quivers of types D and E. A reason for this is that for these quivers there exist short exact sequences σ with indecomposable end terms such that the functions δ σ attain values larger than 1.Proposition 3.5.Let Q be a Dynkin quiver of type D and N = s∈S N s be a decomposition of N ∈ O M (k) such that each representation N s is indecomposable.Let Z p,q ∈ Z 1 M,N (N q , N p ), for p = q in S, be such that the inequality δ σ(N p ,Z p,q ,N q ) ≤ δ M,N does not hold.
Then there is an index r in S \ {p, q} and a homomorphism h q,r in Hom Q (N r , N q ) such that for Y p,r = Z p,q • h q,r the following conditions hold:  ( Proof.We apply Corollary 2.23 for σ = σ(N p , Z p,q , N q ).Hence there is an index r in S and a homomorphism h q,r in Hom Q (N r , N q ) such that for Y p,r = Z p,q • h q,r the following conditions hold: ), and a commutative k-algebra R. Given a representation M in rep(Q), we denote by O M the GL d (k)-orbit in rep d Q (k) which consists of the representations in rep d Q (k) isomorphic to M , where d = dim M .By abuse of notation, we treat O M and its closure O M as reduced subschemes of rep d Q .It is an open and important problem to describe the defining ideal of O M or even to exhibit polynomials having O M as their zero set.If M and N are representations satisfying O N ⊆ O M , then we say that M degenerates to N .Note that O M (k) is the union of O N (k), where N runs through the representations to which M degenerates.Let [X, Y ] = dim k Hom Q (X, Y ), for X, Y ∈ rep(Q).It is well-known that if M degenerates to N then [X, N ] ≥ [X, M ] and [N, X] ≥ [M, X], for any X ∈ rep(Q) (1.1)

Theorem 1 . 3 .
dimension 3, while T N C M has dimension 4 (the trace function does not belong to the ideal I M , see [12, Example 3.7]).Our main result shows that C M and O M have identical tangent spaces in type D: Let Q be a Dynkin quiver of type D and M

Lemma 2 .
22. Let σ : 0 → A α − → B β − → C → 0 be a short exact sequence in rep(Q) and m p,a be any mesh in Z∆.Then δ σ (m p,a ) = 0 if m p,a does not belong to (Γ Q ) 2 .Otherwise, d Q (k[t, t −1 ]).Hence the claim follows from the fact that O M is a GL d -invariant subscheme of rep d Q and by Lemma 3.1 applied to the canonical injective homomorphism k[t] → k[t, t −1 ].

Proposition 3 . 4 .
Let Q be a Dynkin quiver of type A and M ∈ rep(Q).Then O M (k[ε]) = C M (k[ε]).In other words, T N O M = T N C M for any N in O M (k) = C M (k).

( 1 )
The point L = N + Y p,r in rep d Q (k) belongs to O M (k).

( 2 )
N is a proper degeneration of L, i.e.O N O L .In particular, dim O N < dim O L .