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In [3] D. Eisenbud and J. Harris posed the following question: What are the limits of Weierstrass points in families of curves degenerating to stable curves not of compact type? We answer their question for one-dimensional families of smooth curves degenerating to stable curves with just two components meeting at points in general position. In this note we treat only those families whose total space is regular. Nevertheless, we announce here our most general answer, to be presented in detail in [5]. 1. Regularly smoothable linear systems Let C be a connected, projective, nodal curve defined over an algebraically closed field k. Let C1, . . . , Cn be its irreducible components. Let B := Spec(k[[t]]); let o denote its special point and η its generic point. A projective and flat map π : S → B is said to be a smoothing of C if the generic fiber Sη is smooth and the special fiber So is isomorphic to C. In addition, if S is regular then π is called a regular smoothing. If π : S → B is a regular smoothing, then C1, . . . , Cn are Cartier divisors on S, and C1 + · · ·+ Cn ≡ 0. Assume from now on that n = 2. Let ∆ be the reduced Weil divisor with support |∆| = C1 ∩ C2 and δ := deg∆. For i = 1, 2 let gi denote the arithmetic genus and !i the dualizing sheaf of Ci. Then g := g1 + g2 + δ − 1 is the arithmetic genus of C. Let ! denote the dualizing sheaf of C. To avoid exceptional cases, assume from now on that C is semi-stable, that is, assume that δ > 1 or g1g2 > 0. Let li := ⌈g3−i/δ⌉ and mi := liδ− g3−i for i = 1, 2. If l1l2 6= 0 set λi := li/ gcd(l1, l2) for i = 1, 2. Let Li,j := !j((1 + (−1)li)∆) for all i, j ∈ {1, 2}. If π is a regular smoothing of C, let !π be its (relative) dualizing sheaf. Put Lπ,i := !π(−liCi) and Lπ,i := Lπ,i|C for i = 1, 2. Note that Lπ,i|Cj ∼= Li,j for all i, j ∈ {1, 2}. Once we fix isomorphisms we obtain restriction maps ρπ,i,j : H (Lπ,i) → H (Li,j) 1